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1. CJM 2011 (vol 63 pp. 1307)

Dimitrov, Ivan; Penkov, Ivan
A Bott-Borel-Weil Theorem for Diagonal Ind-groups
A diagonal ind-group 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 ind-group and $B\subset G$ is a Borel ind-subgroup, we consider the ind-variety $G/B$ 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 non-generic 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 Bott-Borel-Weil theorem.

Categories:22E65, 20G05

2. 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

3. CJM 2009 (vol 61 pp. 691)

Yu, Xiaoxiang
Prehomogeneity on Quasi-Split 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 non-Archimedean 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 non-abelian, and give necessary and sufficient conditions for $A$ to have a pole at $0$.

Categories:22E50, 20G05

4. 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 non-zero 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.


5. CJM 2002 (vol 54 pp. 1229)

Gow, Roderick; Szechtman, Fernando
The Weil Character of the Unitary Group Associated to a Finite Local Ring
Let $\mathbf{R}/R$ be a quadratic extension of finite, commutative, local and principal rings of odd characteristic. Denote by $\mathbf{U}_n (\mathbf{R})$ the unitary group of rank $n$ associated to $\mathbf{R}/R$. The Weil representation of $\mathbf{U}_n (\mathbf{R})$ is defined and its character is explicitly computed.


6. CJM 2000 (vol 52 pp. 449)

Adler, Jeffrey D.; Roche, Alan
An Intertwining Result for $p$-adic Groups
For a reductive $p$-adic group $G$, we compute the supports of the Hecke algebras for the $K$-types for $G$ lying in a certain frequently-occurring class. When $G$ is classical, we compute the intertwining between any two such $K$-types.

Categories:22E50, 20G05

7. CJM 2000 (vol 52 pp. 438)

Wallach, N. R.; Willenbring, J.
On Some $q$-Analogs of a Theorem of Kostant-Rallis
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 Kostant-Rallis 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 Kostant-Rallis 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

8. 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

9. 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 scheme-theoretic 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 non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model. Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen~\cite{Mehta&vanderKallen} imply a direct-sum 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 direct-sum 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

10. 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

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