1. CJM Online first
 Ovchinnikov, Alexey; Wibmer, Michael

Tannakian categories with semigroup actions
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are
algebraically independent over $\mathbb{C}(x)$. More generally, for
a linear differential or difference equation, it is an important
problem to find all algebraic dependencies among a nonzero solution
$y$ and particular transformations of $y$, such as derivatives
of $y$ with respect to parameters, shifts of the arguments, rescaling,
etc. In the present paper, we develop a theory of Tannakian categories
with semigroup actions, which will be used to attack such questions
in full generality, as each linear differential equation gives
rise to a Tannakian category.
Deligne studied actions of braid groups on categories and obtained
a finite collection of axioms that characterizes such actions
to apply it to various geometric constructions. In this paper,
we find a finite set of axioms that characterizes actions of
semigroups that are finite free products of semigroups of the
form $\mathbb{N}^n\times
\mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$
on Tannakian categories. This is the class of semigroups that
appear in many applications.
Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters Categories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06 

2. CJM 2016 (vol 68 pp. 395)
 Garibaldi, Skip; Nakano, Daniel K.

Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
The representation theory of semisimple algebraic groups over
the complex numbers (equivalently, semisimple complex Lie algebras
or Lie groups, or real compact Lie groups) and the question of
whether a
given complex representation is symplectic or orthogonal has
been solved since at least the 1950s. Similar results for Weyl
modules of split reductive groups over fields of characteristic
different from 2 hold by
using similar proofs. This paper considers analogues of these
results for simple, induced and tilting modules of split reductive
groups over fields of prime characteristic as well as a complete
answer for Weyl modules over fields of characteristic 2.
Keywords:orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups Categories:20G05, 11E39, 11E88, 15A63, 20G15 

3. CJM 2011 (vol 63 pp. 1307)
 Dimitrov, Ivan; Penkov, Ivan

A BottBorelWeil Theorem for Diagonal Indgroups
A diagonal indgroup is a direct limit of classical affine algebraic
groups of growing rank under a class of
inclusions that contains the inclusion
$$
SL(n)\to SL(2n), \quad
M\mapsto \begin{pmatrix}M & 0 \\ 0 & M \end{pmatrix}
$$
as a typical special case. If $G$ is a diagonal indgroup and
$B\subset G$ is a Borel indsubgroup,
we consider the indvariety $G/B$ and compute the cohomology
$H^\ell(G/B,\mathcal{O}_{\lambda})$
of any $G$equivariant line bundle $\mathcal{O}_{\lambda}$ on
$G/B$. It has been known that, for a generic $\lambda$,
all cohomology groups of $\mathcal{O}_{\lambda}$ vanish, and that a
nongeneric equivariant
line bundle $\mathcal{O}_{\lambda}$ has at most one
nonzero cohomology group. The new result of this paper is a
precise description of when
$H^j(G/B,\mathcal{O}_{\lambda})$ is nonzero and the proof of the fact
that, whenever nonzero,
$H^j(G/B, \mathcal{O}_{\lambda})$ is a $G$module dual to a highest
weight module.
The main difficulty is in defining an appropriate analog $W_B$ of the
Weyl group, so that the action of $W_B$
on weights of $G$ is compatible with the analog of the Demazure
``action" of the Weyl group on the cohomology
of line bundles. The highest weight corresponding to $H^j(G/B,
\mathcal{O}_{\lambda})$ is then computed
by a procedure similar to that in the classical BottBorelWeil theorem.
Categories:22E65, 20G05 

4. CJM 2009 (vol 62 pp. 34)
 Campbell, Peter S.; Nevins, Monica

Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$adic Field
We decompose the restriction of ramified principal series
representations of the $p$adic group $\mathrm{GL}(3,\mathrm{k})$ to its
maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is
dependent on the degree of ramification of the inducing characters and
can be characterized in terms of filtrations of the Iwahori subgroup
in $K$. We establish several irreducibility results and illustrate
the decomposition with some examples.
Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$adic groups Categories:20G25, 20G05 

5. CJM 2009 (vol 61 pp. 691)
 Yu, Xiaoxiang

Prehomogeneity on QuasiSplit Classical Groups and Poles of Intertwining Operators
Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit,
connected, reductive classical group $G$ defined over a nonArchimedean
field and $A$ is the standard intertwining operator attached to a
tempered representation of $G$ induced from $M$. In this paper we
determine all the cases in which $\Lie(N)$ is
prehomogeneous under $\Ad(m)$ when $N$ is nonabelian, and give necessary
and sufficient conditions for $A$ to have a pole at $0$.
Categories:22E50, 20G05 

6. CJM 2007 (vol 59 pp. 449)
 Badulescu, Alexandru Ioan

$\SL_n$, Orthogonality Relations and Transfer
Let $\pi$ be a square integrable representation of
$G'=\SL_n(D)$, with $D$ a central division algebra of finite dimension
over a local field $F$ \emph{of nonzero characteristic}. We prove
that, on the elliptic set, the character of $\pi$ equals the complex
conjugate of the orbital integral of one of the pseudocoefficients
of~$\pi$. We prove also the orthogonality relations for characters of
square integrable representations of $G'$. We prove the stable
transfer of orbital integrals between $\SL_n(F)$ and its inner forms.
Category:20G05 

7. CJM 2002 (vol 54 pp. 1229)
8. CJM 2000 (vol 52 pp. 449)
9. CJM 2000 (vol 52 pp. 438)
 Wallach, N. R.; Willenbring, J.

On Some $q$Analogs of a Theorem of KostantRallis
In the first part of this paper generalizations of Hesselink's
$q$analog of Kostant's multiplicity formula for the action of a
semisimple Lie group on the polynomials on its Lie algebra are given
in the context of the KostantRallis theorem. They correspond to the
cases of real semisimple Lie groups with one conjugacy class of Cartan
subgroup. In the second part of the paper a $q$analog of the
KostantRallis theorem is given for the real group $\SL(4,\mathbb{R})$
(that is $\SO(4)$ acting on symmetric $4 \times 4$ matrices). This
example plays two roles. First it contrasts with the examples of the
first part. Second it has implications to the study of entanglement
of mixed 2 qubit states in quantum computation.
Categories:22E47, 20G05 

10. CJM 2000 (vol 52 pp. 265)
 Brion, Michel; Helminck, Aloysius G.

On Orbit Closures of Symmetric Subgroups in Flag Varieties
We study $K$orbits in $G/P$ where $G$ is a complex connected
reductive group, $P \subseteq G$ is a parabolic subgroup, and $K
\subseteq G$ is the fixed point subgroup of an involutive
automorphism $\theta$. Generalizing work of Springer, we
parametrize the (finite) orbit set $K \setminus G \slash P$ and we
determine the isotropy groups. As a consequence, we describe the
closed (resp. affine) orbits in terms of $\theta$stable
(resp. $\theta$split) parabolic subgroups. We also describe the
decomposition of any $(K,P)$double coset in $G$ into
$(K,B)$double cosets, where $B \subseteq P$ is a Borel subgroup.
Finally, for certain $K$orbit closures $X \subseteq G/B$, and for
any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero
global sections, we show that the restriction map $\res_X \colon
H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and
that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we
describe the $K$module $H^0 (X, \mathcal{L})$. This gives
information on the restriction to $K$ of the simple $G$module $H^0
(G/B, \mathcal{L})$. Our construction is a geometric analogue of
Vogan and Sepanski's approach to extremal $K$types.
Keywords:flag variety, symmetric subgroup Categories:14M15, 20G05 

11. CJM 1998 (vol 50 pp. 525)
 Brockman, William; Haiman, Mark

Nilpotent orbit varieties and the atomic decomposition of the $q$Kostka polynomials
We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of
schemetheoretic
intersections of nilpotent orbit closures with the diagonal matrices.
Here $\mu'$ gives the Jordan block structure of the nilpotent matrix.
de Concini and Procesi~\cite{deConcini&Procesi} proved a conjecture of
Kraft~\cite{Kraft} that these rings are isomorphic to the cohomology
rings of the varieties constructed by
Springer~\cite{Springer76,Springer78}. The famous $q$Kostka
polynomial~$\Klmt(q)$ is the Hilbert series for the
multiplicity of the irreducible symmetric group representation indexed
by~$\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$.
\LS~\cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition
of~$\Klmt(q)$ as a sum of ``atomic'' polynomials with
nonnegative integer coefficients, and Lascoux proposed a
corresponding decomposition in the cohomology model.
Our work provides a geometric interpretation of the atomic
decomposition. The Frobeniussplitting results of Mehta and van der
Kallen~\cite{Mehta&vanderKallen} imply a directsum decomposition of
the ideals of nilpotent orbit closures, arising from the inclusions of
the corresponding sets. We carry out the restriction to the diagonal
using a recent theorem of Broer~\cite{Broer}. This gives a directsum
decomposition of the ideals yielding the $k[\Cmubar\cap
\hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of
the $q$Kostka polynomials.
Keywords:$q$Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varieties Categories:05E10, 14M99, 20G05, 05E15 

12. CJM 1997 (vol 49 pp. 133)
 Reeder, Mark

Exterior powers of the adjoint representation
Exterior powers of the adjoint representation of a complex semisimple Lie
algebra are decomposed into irreducible representations, to varying
degrees of satisfaction.
Keywords:Lie algebras, adjoint representation, exterior algebra Categories:20G05, 20C30, 22E10, 22E60 
