496 research outputs found
Differential Calculi on Some Quantum Prehomogeneous Vector Spaces
This paper is devoted to study of differential calculi over quadratic
algebras, which arise in the theory of quantum bounded symmetric domains. We
prove that in the quantum case dimensions of the homogeneous components of the
graded vector spaces of k-forms are the same as in the classical case. This
result is well-known for quantum matrices.
The quadratic algebras, which we consider in the present paper, are
q-analogues of the polynomial algebras on prehomogeneous vector spaces of
commutative parabolic type. This enables us to prove that the de Rham complex
is isomorphic to the dual of a quantum analogue of the generalized
Bernstein-Gelfand-Gelfand resolution.Comment: LaTeX2e, 51 pages; changed conten
On the irreducibility of locally analytic principal series representations
Let G be a p-adic connected reductive group with Lie algebra g. For a
parabolic subgroup P in G and a finite-dimensional locally analytic
representation V of P, we study the induced locally analytic G-representation W
= Ind^G_P(V). Our result is the following criterion concerning the topological
irreducibility of W: if the Verma module U(g) \otimes_{U(p)} V' associated to
the dual representation V' is irreducible then W is topologically irreducible
as well.Comment: 44 pages; final version. An appendix has been added in which it is
shown that the canonical maps between certain completions of distribution
algebras are injective. This fills a gap in a previous version; it was
pointed out to us by a refere
Poset pinball, GKM-compatible subspaces, and Hessenberg varieties
This paper has three main goals. First, we set up a general framework to
address the problem of constructing module bases for the equivariant cohomology
of certain subspaces of GKM spaces. To this end we introduce the notion of a
GKM-compatible subspace of an ambient GKM space. We also discuss
poset-upper-triangularity, a key combinatorial notion in both GKM theory and
more generally in localization theory in equivariant cohomology. With a view
toward other applications, we present parts of our setup in a general algebraic
and combinatorial framework. Second, motivated by our central problem of
building module bases, we introduce a combinatorial game which we dub poset
pinball and illustrate with several examples. Finally, as first applications,
we apply the perspective of GKM-compatible subspaces and poset pinball to
construct explicit and computationally convenient module bases for the
-equivariant cohomology of all Peterson varieties of classical Lie type,
and subregular Springer varieties of Lie type . In addition, in the Springer
case we use our module basis to lift the classical Springer representation on
the ordinary cohomology of subregular Springer varieties to -equivariant
cohomology in Lie type .Comment: 32 pages, 4 figure
On Extensions of generalized Steinberg Representations
Let F be a local non-archimedean field and let G be the group of F-valued
points of a reductive algebraic group over F. In this paper we compute the
Ext-groups of generalized Steinberg representations in the category of smooth
G-representations with coefficients in a certain self-injective ring.Comment: 22 pages, LaTe
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