We establish a duality within the spectral sequence that governs the
holomorphic double fibration transform. It has immediate application to the
questions of injectivity and range characterization for this transform. We
discuss some key examples and an improved duality that holds in the Hermitian
holomorphic case.

The transfer factors for standard endoscopy involve, among other things, the
Langlands-Shelstad splitting invariant. This note introduces a twisted version
of that splitting invariant. The twisted splitting invariant is then used to
define a better twisted factor $\Delta_I$. In addition we correct a sign error
in the definition of twisted transfers. There are two ways to correct the sign
error. One way yields twisted transfer factors $\Delta'$ that are compatible
with the classical Langlands correspondence.

We investigate various topological spaces and varieties which can be
associated to a block of a finite group scheme G. These spaces come from the
theory of cohomological support varieties for modules, as well as from the
representation-theoretic constructions of E. Friedlander and J. Pevtsova.

Let T be a triangulated category with triangulated subcategories X and Y. We
show that the subcategory of extensions X * Y is triangulated if and only if Y
* X is contained in X * Y.

In this situation, we show the following analogue of the Second Isomorphism
Theorem: (X * Y) / X is equivalent to Y / (X \cap Y) and (X * Y) / Y is
equivalent to X / (X \cap Y).

The spin analogues of several classical concepts and results for Hecke
algebras are established. A Frobenius type formula is obtained for irreducible
characters of the Hecke-Clifford algebra. A precise characterization of the
trace functions allows us to define the character table for the algebra. The
algebra is endowed with a canonical symmetrizing trace form, with respect to
which the spin generic degrees are formulated and shown to coincide with the
spin fake degrees.

The nonstandard Hecke algebra \check{\mathscr{H}}_r was defined in GCT IV to
study the Kronecker problem. We study a quotient \check{\mathscr{H}}_{r,2} of
\check{\mathscr{H}}_r, called the nonstandard Temperley-Lieb algebra, which is
simpler than \check{\mathscr{H}}_{r}. It is defined to be the subalgebra of
H_{r,2} \otimes \mathscr{H}_{r,2} generated by \mathcal{P}_i := C'_i \otimes
C'_i + C_i \otimes C_i, i \in [r-1], where \mathscr{H}_{r,2} is the
Temperley-Lieb algebra and C'_i and C_i are Kazhdan-Lusztig basis elements.

Let A be a connected left artinian ring with radical square zero and with n
simple modules. If A is not self-injective, then we show that any module M with
Ext^i(M,A) = 0 for 1 \le i \le n + 1 is projective. We also determine the
structure of the artin algebras with radical square zero and n simple modules
which have a non-projective module M such that Ext^i(M,A) = 0 for 1 \le i \le
n.

We use the representation theory of Lie algebras and computational linear
algebra to obtain an explicit formula for the hyperdeterminant of a $3 \times 3
\times 2$ array: a homogeneous polynomial of degree 12 in 18 variables with
16749 monomials and 41 distinct integer coefficients; the monomials belong to
178 orbits under the action of $(S_3 \times S_3 \times S_2) \rtimes S_2$.

We consider a natural basis of the Iwahori fixed vectors in the Whittaker
model of an unramified principal series representation of a split semisimple p-
adic group, indexed by the Weyl group. We show that the elements of this basis
may be computed from one another by applying Demazure-Lusztig operators. The
precise identities involve correction terms, which may be calculated by a
combinatorial algorithm that is identical to the computation of the fibers of
the Bott-Samelson resolution of a Schubert variety.

The purpose of this paper is to generalize Zhu's theorem about characters of
modules over a vertex operator algebra to the setting of a vertex operator
superalgebra whose vectors may have rational, rather than integer, conformal
weights. It turns out that to recover SL(2, Z)-invariance of the characters, it
is necessary to include twisted modules into the discussion. Another new
feature arises in the super-case; the space of conformal blocks is no longer
spanned by the trace functions of Zhu, the twisted trace functions of Dong, Li
and Mason, and their super-analogues.

Let $W$ be a finite Coxeter group. It is well-known that the number of
involutions in $W$ is equal to the sum of the degrees of the irreducible
characters of $W$. Following a suggestion of Lusztig, we show that this
equality is compatible with the decomposition of $W$ into Kazhdan--Lusztig
cells. The proof uses a generalisation of the Frobenius--Schur indicator to
symmetric algebras, which may be of independent interest.

Zuk's criterion give us a condition for a finitely generated group to have
Property(T): the smallest non - zero eigenvalue of Laplace operator
corresponding to the simple random walk on the associated graph have to be
greater than 1/2. We present here two examples that prove that this condition
cannot be improved.

Adjoint functors and projectivization in representation theory of partially
ordered sets are used to generalize the algorithms of differentiation by a
maximal and by a minimal point. Conceptual explanations are given for the
combinatorial construction of the derived set and for the differentiation
functor

We consider a central extension of the sheaf of Lie algebras of maps from a
manifold into a finite-dimensional simple Lie algebra, together with the sheaf
of vector fields. Using vertex algebra methods we construct sheaves of modules
for this sheaf of Lie algebras. Our results extend the work of
Malikov-Schechtman-Vaintrob on the chiral de Rham complex.

The goal of this paper is to study the representation theory of a classical
infinite-dimensional Lie algebra - the Lie algebra of vector fields on an
N-dimensional torus for N > 1. The case N=1 gives a famous Virasoro algebra (or
its centerless version - the Witt algebra). The algebra of vector fields has an
important class of tensor modules parametrized by finite-dimensional modules of
gl(N). Tensor modules can be used in turn to construct bounded irreducible
modules for the vector fields on N+1-dimensional torus, which are the central
objects of our study.

We show that the complexity of the Lie module $\mathrm{Lie}(n)$ in
characteristic $p$ is bounded above by $m$ where $p^m$ is the largest $p$-power
dividing $n$ and, if $n$ is not a $p$-power, is equal to the maximum of the
complexities of $\Lie(p^i)$ for $1 \leq i \leq m$.

Motivated by known examples of global integrals which represent automorphic
L-functions, this paper initiates the study of a certain two-dimensional array
of global integrals attached to any reductive algebraic group, indexed by
maximal parabolic subgroups in one direction and by unipotent conjugacy classes
in the other. Fourier coefficients attached to unipotent classes,
Gelfand-Kirillov dimension of automorphic representations, and an identity
which, empirically, appears to constrain the unfolding process are presented in
detail with examples selected from the exceptional groups.

In this paper we present Affine.m - program for computations in
representation theory of finite-dimensional and affine Lie algebras and
describe implemented algorithms. Algorithms are based upon the properties of
weights and Weyl symmetry. The most important problems for us are the ones,
concerning computation of weight multiplicities in irreducible and Verma
modules, branching of representations and tensor product decomposition. These
problems have numerous applications in physics and we provide some examples of
these applications.

We give a complete classification of the irreducible quasifinite modules for
algebras of the form Vir \otimes A, where Vir is the Virasoro algebra and A is
a Noetherian commutative associative unital algebra over the complex numbers.
It is shown that all such modules are tensor products of generalized evaluation
modules. We also give an explicit sufficient condition for a Verma module of
Vir \otimes A to be reducible. In the case that A is an infinite-dimensional
integral domain, this condition is also necessary.

We determine the dimensions of irreducible representations of the
three-dimensional Sklyanin algebras. This contributes to the study of marginal
deformations of the N=4 super Yang-Mills theory in four dimensions in
supersymmetric string theory. Namely the classification of such representations
is equivalent to determining the vacua of the aforementioned deformed theories.

We prove that the categories of smooth and analytic unitary representations
of Banach--Lie supergroups are well-behaved under restriction functors, in the
sense that the restriction of a representation to an integral subsupergroup is
well-defined. We also prove that the category of analytic representations is
isomorphic to a subcategory of the category of smooth representations. These
facts are needed as a crucial first step to a rigorous treatment of the
analytic theory of unitary representations of Banach--Lie supergroups.

Cayley's hyperdeterminant is a homogeneous polynomial of degree 4 in the 8
entries of a 2 x 2 x 2 array. It is the simplest (nonconstant) polynomial which
is invariant under changes of basis in three directions. We use elementary
facts about representations of the 3-dimensional simple Lie algebra sl_2(C) to
reduce the problem of finding the invariant polynomials for a 2 x 2 x 2 array
to a combinatorial problem on the enumeration of 2 x 2 x 2 arrays with
non-negative integer entries.

Let $U_q(\g)$ be a quantum generalized Kac-Moody algebra and let $V(\Lambda)$
be the integrable highest weight $U_q(\g)$-module with highest weight
$\Lambda$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra
$R^\Lambda$ provides a categorification of $V(\Lambda)$.

In this article we give an algorithm for determining the generators and
relations for the rings of semi-invariant functions on irreducible components
of representation spaces for gentle string algebras. These rings of
semi-invariants turn out to be semigroup rings to which we can associate a
so-called matching graph. Under this association, generators for the semigroup
can be seen by certain walks on this graph, and relations are given by certain
configurations in the graph. This allows us to determine degree bounds for the
generators and relations of these rings.

It is generally believed (and for the most part is probably true) that Lie
theory, in contrast to the characteristic zero case, is insufficient to tackle
the representation theory of algebraic groups over prime characteristic fields.
However, in this paper we show that, for a large and important class of
unipotent algebraic groups (namely the unipotent upper triangular groups
$U_n$), and under a certain hypothesis relating the characteristic $p$ to both
$n$ and the dimension $d$ of a representation (specifically, $p \geq
\text{max}(n,2d)$), Lie theory is completely sufficient to determine t

The finite groups having an indecomposable polynomial invariant whose degree
is at least half of the order of the group are classified. Apart from four
sporadic exceptions these are exactly the groups having a cyclic subgroup of
index at most two. The Noether bound is determined for these groups, and
estimates are given for various other groups as well.

In this expository paper we present an overview of various graphical
categorifications of the Heisenberg algebra and its Fock space representation.
We begin with a discussion of "weak" categorifications via modules for Hecke
algebras and "geometrizations" in terms of the cohomology of the Hilbert
scheme. We then turn our attention to more recent "strong" categorifications
involving planar diagrammatics and derived categories of coherent sheaves on
Hilbert schemes.

EI-categories are a simultaneous generalisation of finite groups and finite
quivers without oriented cycles. It is therefore a natural question to ask for
a characterisation of finite representation type. For special classes of
EI-categories a complete characterisation is obtained using quiver techniques.
For EI-categories with two objects we present a necessary criterion for finite
representation type. The complexity of this classification problem is
illustrated by some examples.

Suppose a finite group acts on a scheme X and a finite-dimensional Lie
algebra g. The associated equivariant map algebra is the Lie algebra of
equivariant regular maps from X to g. The irreducible finite-dimensional
representations of these algebras were classified in previous work with P.
Senesi, where it was shown that they are all tensor products of evaluation
representations and one-dimensional representations.

If $Q$ is a non degenerate quadratic form on ${\bb C}^n$, it is well known
that the differential operators $X=Q(x)$, $Y=Q(\partial)$, and
$H=E+\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebra
isomorphic to ${\go sl}_{2}$.

The $q$-deformed Fock spaces of higher levels were introduced by
Jimbo-Misra-Miwa-Okado. The $q$-decomposition matrix is a transition matrix
from the standard basis to the canonical basis defined by Uglov in the
$q$-deformed Fock space. In this paper, we show that parts of $q$-decomposition
matrices of level $\ell$ coincides with that of level $\ell - 1$ under certain
conditions of multi charge.

We study orbital integrals and invariant eigendistributions for the symmetric
pair (g,h)=(gl(4,R),gl(2,R)*gl(2,R)). Let q=g/h and let N be the set of
nilpotents of q. We first obtain an asymptotic behavior of orbital integrals
around nonzero semisimple elements of q. We study eigendistributions around
such elements and give an explicit basis of eigendistributions on q-N given by
a locally integrable function on q-N.

We define the category of partitioned binary relations and show that it
contains many classical diagram categories, including categories of binary
relations, maps, injective maps, partitions, (oriented) Brauer diagrams and
(oriented) Temperley-Lieb diagrams. We construct a one-parameter deformation of
the category of partitioned binary relations and show that it gives rise to
classical one-parameter deformations of partition, Brauer and Temperley-Lieb
categories.

We study the Segal-Bargmann transform, or the heat transform, $H_t$ for a
compact symmetric space $M=U/K$. We prove that $H_t$ is a unitary isomorphism
$H_t : L^2(M) \to \cH_t (M_\C)$ using representation theory and the restriction
principle. We then show that the Segal-Bargmann transform behaves nicely under
propagation of symmetric spaces.

This paper deals with the problem of factorizing integer powers of the
Laplace operator acting on functions taking values in higher spin
representations. This is a far-reaching generalization of the well-known fact
that the square of the Dirac operator is equal to the Laplace operator. Using
algebraic properties of projections of Stein-Weiss gradients, i.e. generalized
Rarita-Schwinger and twistor operators, we give a sharp upper bound on the
order of polyharmonicity for functions with values in a given representation
with half-integral highest weight.

We study a limit category $M$ constructed from the polynomial representations
of all general linear groups. We construct a $g$-categorification on $M$ in the
sense of Chuang and Rouquier, which categorifies the Fock space representation
of $g$, (here $g$ is either $\hat{sl}_p$ or $sl_{\infty}$ depending on the
characteristic of the ground field). We construct the Misra-Miwa crystal of
Fock space from the set of simple objects of $M$.

This paper is an exposition of W.B. Arveson's complete invariant for the
unitary similarity of complex, irreducible matrices.

In a recent article, G. Malle and G. Navarro conjectured that the $p$-blocks
of a finite group all of whose height 0 characters have the same degree are
exactly the nilpotent blocks defined by M. Brou\'e and L. Puig. In this paper,
we check that this conjecture holds for spin-blocks of the covering group
$2.\A_n$ of the alternating group $\A_n$, thereby solving a case excluded from
the study of quasi-simple groups by Malle and Navarro.

The index of a seaweed Lie algebra can be computed from its associated
meander graph. We examine this graph in several ways with a goal of determining
families of Frobenius (index zero) seaweed algebras. Our analysis gives two new
families of Frobenius seaweed algebras as well as elementary proofs of known
families of such Lie algebras.

Let $A_{n}(S)$ be a coefficient free cluster algebra over a field $K$. A
cluster automorphism is an element of $Aut._{K}K(t_{1}, t_{2},..., t_{n})$
which leaves the set of all cluster variables, $\xi_{S}$, invariant. The group
of all such automorphisms is studied in terms of the orbits of the symmetric
group action on the set of all seeds S and the cluster pattern.

In this paper we give an intimate connection between the characteristic zero
representation theories of the Additive and Heisenberg groups, and their
characteristic p >0 theories when p is much larger than the dimension a
representation. In particular, if p >> dimension, then all characteristic p
representations for these groups can be factored into commuting products of
representations, with each factor arising from a representation of the Lie
algebra of the group, one for each of the representation's Frobenius layers.

Gabriel's Theorem, and the work of Bernstein, Gelfand and Ponomarev
established a connection between the theory of quiver representations and the
theory of simple Lie algebras. Lie superalgebras have been studied from many
perspectives, and many results about Lie algebras have analogues for Lie
superalgebras.

We compute the Hochschild Cohomology of a finite-dimensional preprojective
algebra of generalized Dynkin type Ln over a field of odd characteristic not
dividing 2n+1. This turns out to be periodic by a result of
Bialkowski-Erdmann-Skowro\'{n}ski. In particular, we describe the ring
structure of the Hoschschild Cohomology ring under the Yoneda product by giving
an explicit presentation by generators and relations.

Approximate algebraic structures play a defining role in arithmetic
combinatorics and have found remarkable applications to basic questions in
number theory and pseudorandomness. Here we study approximate representations
of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is
large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2$ is small, where x
and y are uniformly random elements of the group G and U_d denotes the unitary
group of degree d.

Given a Lax system of equations with the spectral parameter on a Riemann
surface we construct a projective unitary representation of the Lie algebra of
Hamiltonian vector fields by Knizhnik-Zamolodchikov operators. This provides a
prequantization of the Lax system. The representation operators of Poisson
commuting Hamiltonians of the Lax system projectively commute.

We present an explicit difference operator diagonalized by the Macdonald
polynomials associated with an (arbitrary) admissible pair of irreducible
reduced crystallographic root systems. By the duality symmetry, this gives rise
to an explicit Pieri formula for the Macdonald polynomials in question. The
simplest examples of our construction recover Macdonald's celebrated difference
operators and associated Pieri formulas pertaining to the minuscule and
quasi-minuscule weights.

We show that with respect to the q-Plancherel measure on partitions of size
n, the irreducible characters of an Hecke algebra $H_q(S_n)$ are concentrated
around the normalized trace of $H_q(S_n)$. More precisely, we prove that the
deviations of the values of the q-characters $\chi^\lambda_q$ are
asymptotically gaussian, and we give an explicit formula for the covariances of
the limit normal laws (the other results were already in arXiv:1001.2180).

A new calculus of planar diagrams involving diagrammatics for biadjoint
functors and degenerate affine Hecke algebras is introduced. The calculus leads
to an additive monoidal category whose Grothendieck ring contains an integral
form of the Heisenberg algebra in infinitely many variables. We construct bases
of vector spaces of morphisms between products of generating objects in this
category.

We extend our previous study of quaternionic analysis based on representation
theory to the case of split quaternions H_R. The special role of the unit
sphere in the classical quaternions H identified with the group SU(2) is now
played by the group SL(2,R) realized by the unit quaternions in H_R. As in the
previous work, we use an analogue of the Cayley transform to relate the
analysis on SL(2,R) to the analysis on the imaginary Lobachevski space
SL(2,C)/SL(2,R) identified with the one-sheeted hyperboloid in the Minkowski
space M.

For an arbitrary unimodular Lie group $G$, we construct strongly continuous
unitary representations in the Bergman space of a naturally constructed
strongly pseudoconvex neighborhood of $G$ in the complexification of its
underlying manifold.

Let $G$ be an arbitrary finite group. The McKay conjecture asserts that $G$
and the normaliser $N_G (P)$ of a Sylow $p$-subgroup $P$ in $G$ have the same
number of characters of degree not divisible by $p$ (that is, of $p'$-degree).
We propose a new refinement of the McKay conjecture, which suggests that one
may choose a correspondence between the characters of $p'$-degree of $G$ and
$N_G (P)$ to be compatible with induction and restriction in a certain sense.
This refinement implies, in particular, a conjecture of Isaacs and Navarro.

Let G be a compact connected semisimple Lie group and let H\subset G be a
closed connected subgroup such that rank(G)=rank(H) and G/H is a symmetric
space. Given an irreducible representation of H, we define a Dirac operator D
and determine the representations of G in the kernel of D. Moreover, we show
that any irreducible representation of G can be constructed in this way. Our
approach is similar to that of Parthasarathy.

It is shown that every nonsingular continuous representation of the group
algebra $L^{1}(G)$ in Banach spaces is completely reducible if and only if $G$
is a compact group.

Let $U_n(q)$ denote the upper triangular group of degree $n$ over the finite
field $\F_q$ with $q$ elements. It is known that irreducible constituents of
supercharacters partition the set of all irreducible characters $Irr(U_n(q)).$
In this paper we present a correspondence between supercharacters and pattern
subgroups of the form $U_k(q)\cap {}^wU_k(q)$ where $w$ is a monomial matrix in
$GL_k(q)$ for some $k<n.$

We define series of representations of the Thompson's groups $F$ and $T$. We
show that they are irreducible and classify them up to unitary equivalence.

We give a proof of a conjecture of Lehrer and Shoji regarding the occurrences
of the exterior powers of the reflection representation in the cohomology of
Springer fibers. The actual theorem proved is a slight extension of the
original conjecture to all nilpotent orbits and also takes into account the
action of the component group. The method is to use Shoji's approach to the
orthogonality formulas for Green functions to relate the symmetric algebra to a
sum over Green functions.

In this note, we point out an error in the proof of Theorem 4.7 of [P. Achar
and A.~Henderson, `Orbit closures in the enhanced nilpotent cone', Adv. Math.
219 (2008), 27-62], a statement about the existence of affine pavings for
fibres of a certain resolution of singularities of an enhanced nilpotent orbit
closure. We also give independent proofs of later results that depend on that
statement, so all other results of that paper remain valid.

We prove the Weyl denominator identity for the affine Lie superalgebra
gl(2|2)^ conjectured by V. Kac and M. Wakimoto. As it was pointed out in their
paper, this gives another proof of Jacobi identity for the number of
presentations of a given integer as a sum of eight squares.

Let $N$ be the Heisenberg group. We consider left-invariant multiplicity free
subspaces of $L^2(N)$. We prove a necessary and sufficient density condition in
order that such subspaces possess the interpolation property with respect to a
class of discrete subsets of $N$ that includes the integer lattice. We exhibit
a concrete example of a subspace that has interpolation for the integer
lattice, and we also prove a necessary and sufficient condition for shift
invariant subspaces to possess a singly-generated orthonormal basis of
translates.

In this paper, we consider certain elements in von Neumann algebras generated
by graph groupoids. In particular, we are interested in finitely supported
elements, called graph operators. We study the characterizations for
self-adjointness, the unitary property, hyponormality and normality of graph
operators.

We give a new bijective interpretation of the Cauchy identity for Schur
operators which is a commutation relation between two formal power series with
operator coefficients. We introduce a plactic algebra associated with the
Kashiwara's extremal weight crystals over the Kac-Moody algebra of type
$A_{+\infty}$, and construct a Knuth type correspondence preserving the plactic
relations.

We give a new short proof of Skowronski and Weyman's theorem about the
structure of the algebras of semi-invariants of Euclidean quivers, in the case
of quivers without oriented cycles. Our proof is based essentially on Derksen
and Weyman's result about the generators of these algebras and properties of
Schofield semi-invariants.

We interpret the Rahman polynomials in terms of the Lie algebra $sl_3(C)$.
Using the parameters of the polynomials we define two Cartan subalgebras for
$sl_3(C)$, denoted $H$ and $\tilde{H}$. We display an antiautomorphism
$\dagger$ of $sl_3(C)$ that fixes each element of $H$ and each element of
$\tilde{H}$.

This is my PhD thesis supervised by Professor Jerzy Weyman. A symmetric
quiver $(Q,\sigma)$ is a finite quiver without oriented cycles $Q=(Q_0,Q_1)$
equipped with a contravariant involution $\sigma$ on $Q_0\sqcup Q_1$. The
involution allows us to define a nondegenerate bilinear form $<,>$ on a
representation $V$ of $Q$. We shall say that $V$ is orthogonal if $<,>$ is
symmetric and symplectic if $<,>$ is skew-symmetric. Moreover we define an
action of products of classical groups on the space of orthogonal
representations and on the space of symplectic representations.

We discuss groups corresponding to Kohno Lie algebra of infinitesimal braids
and actions of such groups. We construct homomorphisms of Lie braid groups to
the group of symplectomorphisms of the space of point configurations in $R^3$
and to groups of symplectomorphisms of coadjoint orbits of $SU(n)$.

In this paper, we give a construction of certain induced complementary series
of the universal covering of the symplectic groups. There are abundant such
representations besides those of linear groups. We achieve this by applying
invariant tensor product to degenerate complementary series on the Shilov
boundary.

Consider the representations of an algebraic group G. In general, polynomial
invariant functions may fail to separate orbits. The invariant subring may not
be finitely generated, or the number and complexity of the generators may grow
rapidly with the size of the representation. We instead study "constructible"
functions defined by straight line programs in the polynomial ring, with a new
"quasi-inverse" that computes the inverse of a function where defined. We write
straight line programs defining constructible functions that separate the
orbits of G.

We prove the Plancherel formula for Whittaker functions on a reductive p-adic
group. This a sequel to our work on Paley-Wiener theorem. Our proof is close to
the proof written by Waldspurger of the Harish-Chandra Plancherel formula for
smooth functions on the group and use many of his results. One simplification
is the easy proof of the Fourier transfom, which follows from a result of
Joseph Bernstein.

For certain Lie algebras g, we can use a Z/5Z-grading and define a quartic
form and a skew-symmetric bilinear form on the degree 1 component, g_1, thereby
constructing a Freudenthal triple system. The structure of the Freudenthal
triple system is examined using root system methods available in the Lie
algebra context. In the cases g = E_8 (where g_1 is the minuscule
representation of E_7) and g = D_4, we determine the groups stabilizing the
quartic form and both the quartic and bilinear forms.

This paper studies self-injective algebras of polynomial growth. We prove
that the derived equivalence classification of weakly symmetric algebras of
domestic type coincides with the classification up to stable equivalences (of
Morita type). As for weakly symmetric non-domestic algebras of polynomial
growth, up to some scalar problems, the derived equivalence classification
coincides with the classification up to stable equivalences of Morita type.

Let $U_\zeta$ be the quantum group (Lusztig form) associated to the simple
Lie algebra $\mathfrak{g}$, with parameter $\zeta$ specialized to an $\ell$-th
root of unity in a field of characteristic $p>0$. In this paper we study
certain finite-dimensional normal Hopf subalgebras $U_\zeta(G_r)$ of $U_\zeta$,
called Frobenius-Lusztig kernels, which generalize the Frobenius kernels $G_r$
of an algebraic group $G$. When $r=0$, the algebras studied here reduce to the
small quantum group introduced by Lusztig. We classify the irreducible
$U_\zeta(G_r)$-modules and discuss their characters.

We study the genuine part of the Arthur-Selberg trace formula for some
nonlinear covers of connected reductive groups. As a first step towards the
invariant trace formula, we express the geometric side in terms of weighted
orbital integrals. In particular, our results apply to the covers constructed
by Brylinksi and Deligne.

Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the
corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with
Cartan datum given by Q, and let W be its Weyl group. With w in W is associated
a unipotent cell N^w of the Kac-Moody group with Lie algebra g. In previous
work we proved that the coordinate ring \C[N^w] of N^w is a cluster algebra in
a natural way.

Let G be a complex connected reductive group. The PRV conjecture, which was
proved independently by S. Kumar and O. Mathieu in 1989, gives explicit
irreducible submodules of the tensor product of two irreducible G-modules. This
paper has three aims. First, we simplify the proof of the PRV conjecture, then
we generalize it to other branching problems. Finally, we find other
irreducible components of the tensor product of two irreducible G-modules that
appear for "the same reason" as the PRV ones.

For a symmetric pair of Lie superalgebras which is strongly reductive and of
even type, we introduce the graded Harish-Chandra homomorphism and,
generalising results of Harish-Chandra, V. Kac, and M. Gorelik, show that its
image coincides with the image of Chevalley's restriction map on symmetric
invariants. The latter is known by the graded generalisation of Chevalley's
restriction theorem that we have obtained jointly with J. Hilgert and M.R.
Zirnbauer.

These are notes for a lecture series given at the Fields Institute Summer
School in Geometric Representation Theory and Extended Affine Lie Algebras,
held at the University of Ottawa in June 2009. We give an introduction to the
geometric realization of crystal graphs via the quiver varieties of Lusztig and
Nakajima. The emphasis is on motivating the constructions through concrete
examples. The relation between the geometric construction of crystals and
combinatorial realizations using Young tableaux is also discussed.

For the affine Hecke algebra of type A at roots of unity, we make explicit
the correspondence between geometrically constructed simple modules and
combinatorially constructed simple modules and prove the modular branching
rule. The latter generalizes work by Vazirani.

In this paper, we prove that a refinement of the Alperin-McKay Conjecture for
$p$-blocks of finite groups, formulated by I. M. Isaacs and G. Navarro in 2002,
holds for all covering groups of the symmetric and alternating groups, whenever
$p$ is an odd prime.

In this paper, it is proved that all irreducible Harish-Chandra modules over
the $\Q$ Heisenberg-Virasoro algebra are of intermediate series (all weight
spaces are 1-dimensional).

In this article, we study the elements with disconnected centralizer in the
Brauer complex associated to a simple algebraic group G defined over a finite
field with corresponding Frobenius map F and derive the number of F-stable
semisimple classes of G with disconnected centralizer when the order of the
fundamental group has prime order. We also discuss extendibility of semisimple
characters to their inertia group in the full automorphism group.

We verify Dade's invariant conjecture for the simple Ree groups 2F4(2^{2n+1})
for all n > 0 in the defining characteristic, i.e., in characteristic 2. This
completes the proof of Dade's conjecture for the simple Ree groups
2F4(2^{2n+1}).

We give two integrability criteria for representations of Banach--Lie
algebras as skew-symmetric unbounded operators on a dense domain of a Hilbert
space.

Let $\g$ be a classical Lie algebra, $e$ a nilpotent of $\g$ element and $\gt
g_e$ the centraliser of $e$ in $\g$. We prove that $\g_e=[\g_e,\g_e]$ if and
only if $e$ is rigid. It is also shown that if $e$ is contained in
$[\g_e,\g_e]$, then the nilpotent radical of $\g_e$ coincides with
$[\g(1)_e,\g_e]$, where $\g(1)_e$ is an eigenspace of a characteristic of $e$
with the eigenvalue 1.

We examine the commuting elements $\theta_i=\sum_{j\neq i}
\frac{s_{ij}}{z_i-z_j}$, $z_i\neq z_j$, $s_{ij}$ the transposition swapping $i$
and $j$, and we study their actions on irreducible $S_n$ representations. By
applying Schur-Weyl duality to the results of \cite{RV:QuasiKZ}, we establish a
Bethe Ansatz for these operators which yields joint eigenvectors for each
critical point of a master function.

We discuss some connections between the closure $\bar F$ of a Steinberg fiber
in the wonderful compactification of an adjoint group and the affine
Deligne-Lusztig varieties $X_w(1)$ in the affine flag variety. Among other
things, we describe the emptiness/nonemptiness pattern of $X_w(1)$ if the
translation part of $w$ is quasi-regular. As a by-product, we give a new proof
of the explicit description of $\bar F$, first obtained in \cite{H1}.

We study the first extension groups between Verma modules. There was a
conjecture which claims that the dimensions of the higher extension groups
between Verma modules are the coefficients of $R$-polynomials defined by
Kazhdan-Lusztig. This conjecture was known as the Gabber-Joseph conjecture
(although Gebber and Joseph did not state.) However, Boe gives a counterexample
to this conjecture. In this paper, we study how far are the dimensions of
extension groups from the coefficients of $R$-polynomials.

We decompose the fibers of the Springer resolution for the odd nilcone of the
Lie superalgebra $\osp(2n+1,2n)$ into locally closed subsets. We use this
decomposition to prove that almost all fibers are connected. However, in
contrast with the classical Springer fibers, we prove that the fibers can be
disconnected and non equidimensional.

We use the rational tableaux introduced by Stembridge to give a bideterminant
basis for a normal reductive monoid and for its variety of noninvertible
elements. We also obtain a bideterminant basis for the full coordinate ring of
the general linear group and for all its truncations with respect to saturated
sets. Finally, we deduce an alternative proof of the double centraliser theorem
for the rational Schur algebra and the walled Brauer algebra over an arbitrary
infinite base field which was first obtained by Dipper, Doty and Stoll.

This article studies cluster-tilted algebras whose quiver is cyclically
oriented. In this case an explicit description of the defining relations is
given. For this kind of algebras, it is also shown that there exists an
admissible cut and moreover that each arrow of the quiver is contained in an
admissible cut. Furthermore, we show that if the endomorphism ring of an
algebra of global dimension two over its cluster category, in the sense of
Amiot, is cluster-tilted and has a cyclically oriented quiver, then the
original algebra is a quotient by an admissible cut.

Let D(e) denote the weighted Dynkin diagram of a nilpotent element $e$ in
complex simple Lie algebra $\g$. We say that D(e) is divisible if D(e)/2 is
again a weighted Dynkin diagram. (That is, a necessary condition for
divisibility is that $e$ is even.) The corresponding pair of nilpotent orbits
is said to be friendly. In this note, we classify the friendly pairs and
describe some of their properties. We also observe that any subalgebra sl(3) in
$\g$ determines a friendly pair. Such pairs are called A2-pairs.

Let $\Delta$ be a quiver and let $P$ be a projective representation of
$\Delta$. We study the adjoint action of $Aut P$ on the space of radical
endomorphisms and show that there is a dense open orbit for all $P$, if and
only if $\Delta$ is a Dynkin quiver. Our main application is to show the
existence of dense open orbits for the adjoint action of subgroups of
parabolics in $Gl_n$ on corresponding subalgebras.

We consider the socle deformations arising from formal deformations of a
class of Koszul self-injective special biserial algebras which occur in the
study of the Drinfeld double of the generalized Taft algebras. We show, for
these deformations, that the Hochschild cohomology ring modulo nilpotence is a
finitely generated commutative algebra of Krull dimension 2.

In this article a general framework for studying analytic representations of
a real Lie group G is introduced. Fundamental topological properties of the
representations are analyzed. A notion of temperedness for analytic
representations is introduced, which indicates the existence of an action of a
certain natural algebra A(G) of analytic functions of rapid decay.

In this paper the author finds explicitly all finite-dimensional irreducible
representations of a series of finite permutation groups that are homomorphic
images of Artin braid group.

In this paper we investigate the endomorphism algebras of standard cluster
tilting objects in the stably 2-Calabi-Yau categories $\Sub{\Lambda_w}$ with
elements $w$ in Coxeter groups in \cite{BIRSc}. They are examples of the
2-Auslander algebras introduced in \cite{I1}. Generalizing work in \cite{GLS1}
we show that they are quasihereditary, even strongly quasihereditary in the
sense of \cite{R}.

The most degenerate unitary principal series representations
$\pi_{i\lambda,\delta}$ of $G = GL(N,\R)$ attain the minimum of the
Gelfand--Kirillov dimension among all irreducible unitary representations of
$G$. This article gives an explicit formula of the irreducible decomposition of
the restriction $\pi_{i\lambda,\delta}|_H$ (\textit{branching law}) with
respect to any symmetric pair $(G,H)$. In particular, we prove that the
restriction $\pi_{i\lambda,\delta}|_H$ is always irreducible for $H=Sp(n,\R)$
if $N=2n$ and $n \ge 2$.

In this paper we show that any irreducible finite dimensional representation
of $SL_{n+1}$ remains indecomposable if restricted to n--dimensional abelian
subalgebras spanned by simple root vectors.

We use coefficient systems on the affine Bruhat-Tits building to study
admissible representations of reductive p-adic groups in characteristic not
equal to p. We show that the character function is locally constant and provide
explicit neighbourhoods of constancy. We estimate the growth of the subspaces
of invariants for compact open subgroups.

We construct irreducible modules for twisted toroidal Lie algebras and
extended affine Lie algebras. This is done by combining the representation
theory of untwisted toroidal algebras with the technique of thin coverings of
modules. We illustrate our method with examples of extended affine Lie algebras
of Clifford type.

In this paper, we study the relationship between the McKay quivers of a
finite subgroups $G$ of special linear groups general linear groups, via some
natural extension and embedding. We show that the McKay quiver of certain
extension of a finite subgroup $G$ of $\mathrm{SL}(m,\mathbb C)$ in
$\mathrm{GL}(m,\mathbb C)$ is a regular covering of the McKay quiver of $G$,
and when embedding $G$ in a canonical way into $\mathrm{GL}(m-1,\mathbb C)$,
the new McKay quiver is obtained by adding an arrow from the Nakayama
translation of $i$ back to $i$ for each $i$.

The quantum loop algebra $U_{v}(\mathcal{L}\mathfrak{g})$ was defined as a
generalization of the Drinfeld's new realization of quantum affine algebra to
the loop algebra of any Kac-Moody algebra $\mathfrak{g}$. Schiffmann \cite{S}
has proved (and conjectured) that the Hall algebra of the category of coherent
sheaves over weighted projective lines provides a realization of
$U_{v}(\mathcal{L}\mathfrak{g})$ for those $\mathfrak{g}$ associated to a
star-shaped Dynkin diagram.

The crystal base of the modified quantum group of type $A_{+\infty}$ is
realized as a set of integral bimatrices. It is obtained by describing
explicitly the tensor product of a highest weight crystal and a lowest weight
crystal, and then its limit using a tableaux model of extremal weight crystals.
This realization induces a bicrystal structure of the crystal base of the
modified quantum group and hence its Peter-Weyl type decomposition in a purely
combinatorial way generalizing the classical RSK correspondence.

The irreducible representations of the Witt algebra $W$ are completely known.
A classification of the irreducible $U_\chi(W)$--modules was first established
by Chang and later simplified by Strade. The aim of this article is to give a
classification of the extensions of the simple $U_\chi(W)$--modules having
$p$--character $\chi$ of height $-1,0,1$ and $p-1$ where $p$ denotes the
characteristic of the ground field.

We find the Fisher decomposition for the space of polynomials valued in the
Segal-Shale-Weil representation. As a consequence, this allows to determine
symplectic monogenics, i.e. the space of polynomial solutions of the symplectic
Dirac operator.

We consider the algebra of invariants of binary forms of degree 9 with
complex coefficients, find the 92 basic invariants, give an explicit system of
parameters and show the existence of four more systems of parameters with
different sets of degrees.

We study the PBW filtration on the $sl_{n+1}$ highest weight representations
$V$ of highest weight $\lambda$. This filtration is induced by the standard
degree filtration on $U(n^-)$. We give a description of the associated graded
$S(n^-)$-module $gr V$ in terms of generators and relations. We also construct
a basis of this module. As an application we obtain a new class of bases of the
modules $V$ and derive a graded combinatorial character formula for $V$.

In this note we give representations for the partition algebra A_3(Q) in
Young's seminormal form. For this purpose, we also give characterizations of
A_n(Q) and$A_{n-1/2}(Q).

We formulate a theory of invariants for the spin symmetric group in some
suitable modules which involve the polynomial and exterior algebras. We solve
the corresponding graded multiplicity problem in terms of specializations of
the Schur Q-functions and a shifted q-hook formula. In addition, we provide a
bijective proof for a formula of the principal specialization of the Schur
Q-functions.

This paper is concerned with the values of Harish-Chandra characters of a
class of positive-depth, toral, very supercuspidal representations of $p$-adic
symplectic and special orthogonal groups, near the identity element. We declare
two representations equivalent if their characters coincide on a specific
neighbourhood of the identity (which is larger than the neighbourhood on which
Harish-Chandra local character expansion holds).

Let $Q$ be a wild $n$-Kronecker quiver, i.e., a quiver with two vertices,
labeled by 1 and 2, and $n\geq 3$ arrows from 2 to 1. The indecomposable
regular modules with preprojective Gabriel-Roiter submodules, in particular,
those $\tau^{-i}X$ with $\udim X=(1,c)$ and $1\leq c\leq n-1$ will be studied.
It will be shown that for each $i\geq 1$ the irreducible monomorphisms starting
with $\tau^{-i}X$ are namely Gabriel-Roiter inclusions, and moreover, the
Gabriel-Roiter measures of these modules are `neighbors'.

Y. Palu has generalized the cluster multiplication formulas to 2-Calabi-Yau
categories with cluster tilting objects (\cite{Palu2}). The aim of this note is
to construct a variant of Y. Palu's formula and deduce a new version of the
cluster multiplication formula (\cite{XiaoXu}) for acyclic quivers in the
context of cluster categories.

We address two fundamental questions in the representation theory of affine
Hecke algebras of classical types. One is an inductive formula for
$W$-characters of tempered modules, and the other is the determination of the
constants in the formal degrees of discrete series (in the form conjectured by
Reeder \cite{Re}). The former is completely different than the Lusztig-Shoji
algorithm \cite{Sh, L}, and it is more effective in a number of cases.

Vershik and Kerov conjectured in 1985 that suitably normalized dimensions of
irreducible representations of finite symmetric groups converge to a constant
with respect to the Plancherel family of measures on the space of Young
diagrams. The main result of this paper is the proof of the Vershik-Kerov
conjecture. The argument is based on the methods of Borodin, Okounkov and
Olshanski.

Let $G$ be a connected semisimple algebraic group over an algebraically
closed field of characteristic zero, and let $\th$ be an automorphism of $G$.
We give a characterization of $\th$-twisted spherical conjugacy classes in $G$
by a formula for their dimensions in terms of certain elements in the Weyl
group of $G$, generalizing a result of N. Cantarini, G. Carnovale, and M.
Costantini when $\th$ is the identity automorphism. For $G$ simple and $\th$ an
outer automorphism of $G$, we also classify the Weyl group elements that appear
in the dimension formula.

For a connected complex semi-simple Lie group $G$ and a fixed pair $(B, B^-)$
of opposite Borel subgroups of $G$, we determine when the intersection of a
conjugacy class $C$ in $G$ and a double coset $BwB^-$ is non-empty, where $w$
is in the Weyl group $W$ of $G$. The question comes from Poisson geometry, and
our answer is in terms of the Bruhat order on $W$ and an involution $\mc \in W$
associated to $C$. We study properties of the elements $\mc$.

We prove a conjecture of the first author for $GL_2(F)$, where $F$ is a
finite extension of $Q_p$.

Orthosymplectic Lie superalgebras are fundamental symmetries in modern
physics, such as massive supergravity. However, their representations are far
from being thoroughly understood. In the present paper, we completely determine
the structure of their various supersymmetric polynomial representations
obtained by swapping bosonic multiplication operators and differential
operators in the canonical supersymmetric polynomial representations.

Let $H^*(\calB_e)$ be the cohomology of the Springer fibre for the nilpotent
element $e$ in a simple Lie algebra $\g$, on which the Weyl group $W$ acts by
the Springer representation. Let $\Lambda^i V$ denote the $i$th exterior power
of the reflection representation of $W$. We determine the degrees in which
$\Lambda^i V$ occurs in the graded representation $H^*(\calB_e)$, under the
assumption that $e$ is regular in a Levi subalgebra and satisfies a certain
extra condition which holds automatically if $\g$ is of type A, B, or C.

Given a quiver, a fixed dimension vector, and a positive integer n, we
construct a functor from the category of D-modules on the space of
representations of the quiver to the category of modules over a corresponding
Gan-Ginzburg algebra of rank n. When the quiver is affine Dynkin we obtain an
explicit construction of representations of the corresponding wreath-product
symplectic reflection algebra of rank n.

We show that the algebras describing blocks of the category of cuspidal
weight (respectively generalized weight) $\mathfrak{sl}_n$-modules are
one-parameter (respectively multi-parameter) deformations of certain Brauer
tree algebras. We explicitly determine these deformations both graded and
ungraded. The algebras we deform also appear as special centralizer subalgebras
of Temperley-Lieb algebras or as generalized Khovanov algebras.

In this paper, we are interested in the asymptotic size of rows and columns
of a random Young diagram under a natural deformation of the Plancherel measure
coming from Hecke algebras. The first lines of such diagrams are typically of
order $n$, so it does not fit in the context studied by P. Biane and P.
\'Sniady. Using the theory of polynomial functions on Young diagrams of Kerov
and Olshanski, we are able to compute explicitly the first- and second-order
asymptotics of the length of the first rows.

We prove the periodicity conjecture for pairs of Dynkin diagrams using
Fomin-Zelevinsky's cluster algebras and their (additive) categorification via
triangulated categories.

We consider the general notion of coloured quiver mutation and show that the
mutation class of a coloured quiver $Q$, arising from an $m$-cluster tilting
object associated with $H$, is finite if and only if $H$ is of finite or tame
representation type, or it has at most 2 simples. This generalizes a result
known for 1-cluster categories.

We explicitly calculate a projective bimodule resolution for a special
biserial algebra giving rise to the Hecke algebra H_q(S_4) when q=-1. We then
determine the dimensions of the Hochschild cohomology groups.

A Jantzen type filtration for generalised Varma modules of Lie superalgebras
is introduced. In the case of type I Lie superalgebras, it is shown that the
generalised Jantzen filtration for any Kac module is the unique Loewy
filtration, and the decomposition numbers of the layers of the filtration are
determined by the coefficients of inverse Kazhdan-Lusztig polynomials.
Furthermore, the length of the Jantzen filtration for any Kac module is
determined explicitly in terms of the degree of atypicality of the highest
weight.

We prove the local Gross-Prasad conjecture for generic L-packets of
representations of special orthogonal groups. The proof uses the same result
for tempered L-packets proved in a preceding paper, and irreducibility results
for the induced representations of whose the elements of the L-packets are
Langlands quotients.

Let G be a second countable, locally compact group and let f be a continuous
Herz-Schur multiplier on G. Our main result gives the existence of a (not
necessarily uniformly bounded) strongly continuous representation on a Hilbert
space, such that f is the coefficient of this representation with respect to
two vectors with bounded orbit. Moreover, we show that the norm of the
representation of an element g from G is at most exponential in terms of the
metric distance from g to the identity element of G.

The rational Chow ring A?(S[n],Q) of the Hilbert scheme S[n] parametrising
the length n zero-dimensional subschemes of a toric surface S can be described
with the help of equivariant techniques. In this paper, we explain the general
method and we illustrate it through many examples. In the last section, we
present results on the intersection theory of graded Hilbert schemes.

Let G be a p-adic Lie group. This paper is about the Jordan-Hoelder series of
locally analytic G-representations which are induced from locally algebraic
representations of a parabolic subgroup.

Small representations of a group bring us to large symmetries in a
representation space. Analysis on minimal representations utilises large
symmetries in their geometric models, and serves as a driving force in creating
new interesting problems that interact with other branches of mathematics.

This article discusses the following three topics that arise from minimal
representations of the indefinite orthogonal group:

1. construction of conservative quantities for ultra-hyperbolic equations,

2. quantative discrete branching laws,