Abstract. We study and generalize the connection between the phylogenetic Hilbert functions of Buczynska and Wisniewski [BW] and the Verlinde formula, as discovered by Sturmfels and Xu in [StX]. In order to accomplish this we introduce deformations of algebras of non-abelian theta functions for a general simple complex Lie algebra g structured on the moduli stack of stable punctured curves. We ...

Let G → GL(n,C) be a representation of a complex reductive group. A theorem of Hilbert says that the algebra C[x1, . . . , xn] of invariant polynomials is finitely generated. Let Y be the projective variety defined by this graded algebra, and then we have a rational morphism π : C · · · → Y(C). A theorem on geometric reductivity of Mumford says that a point x ∈ C is regular for the map π if and...

Let F be a number field, G the general linear group of degree n defined over F. Let π be an irreducible cuspidal automorphic representation of G(A). In [1–3], a Rankin-Selbergtype integral is constructed to represent the L function of π. That the integrals of Jacquet, Piatetski-Shapiro, and Shalika are Eulerian follows from the uniqueness of Whittaker models and the fact that cuspidal represent...

We described in [7] a monoid b G acting on the integrable highest weight modules of a symmetrizable Kac-Moody algebra. It has similar structural properties as a reductive algebraic monoid with unit group a Kac-Moody group G. Now we find natural extensions of the action of the Kac-Moody group G on its building Ω to actions of the monoid b G on Ω. These extensions are partly motivated by represen...

Write E for the stable discrete series character associated to an irreducible nite-dimensional representation E of a connected real reductive group G. Let M be the centralizer of the split component of a maximal torus T , and denote by M ( ; E) Arthur's extension of jDG M ( )j 12 E( ) to T (R). In this paper we give a simple explicit expression for M ( ; E), when is elliptic in G. We do not ass...

We obtain a formula for the values of the characteristic function of a character sheaf, in terms of the representation theory of certain finite groups related to the Weyl group. This formula, a generalization of previous results due to Mœglin and Waldspurger, depends on knowledge of certain reductive subgroups that admit cuspidal character sheaves. For quasi-simple groups, we make the formula t...

Let k be an algebraically closed field of characteristic > 2, F = k((t)) and G = Sp 2d . In this paper we propose a geometric analog of the Weil representation of the metaplectic group G̃(F ). This is a category of perverse sheaves on some stack, on which G̃(F ) acts by functors. This construction will be used in [5] (and subsequent publications) for the proof of the geometric Langlands functoria...

An infinite-dimensional representation π of a real reductive Lie group G can often be thought of as a function space on some manifold X. Although X is not uniquely defined by π, there are “geometric invariants” of π, first introduced by Roger Howe in the 1970s, related to the geometry of X. These invariants are easy to define but difficult to compute. I will describe some of the invariants, and...

Many fundamental questions in theoretical computer science are naturally expressed as special cases of the following problem: Let G be a complex reductive group, let V G-module, and v,w elements V. Determine if w is G-orbit closure v. I explain problems, representation theory algebraic geometry that they give rise to, new perspectives on old areas such invariant have arisen light these question...