If q is an algebraic Lie algebra and Q is an algebraic group with Lie algebra q, then ind q equals the transcendence degree of the field of Q-invariant rational functions on q. If q is reductive, then q and q are isomorphic as q-modules and hence ind q = rk q. It is an important invariant-theoretic problem to study index and, more generally, the coadjoint representation for non-reductive Lie al...

The Bernstein center of a reductive p-adic group is the algebra of conjugation invariant distributions on the group which are essentially compact, i.e., invariant distributions whose convolution against a locally constant compactly supported function is again locally constant compactly supported. The center acts naturally on any smooth representation, and if the representation is irreducible, e...

In this paper, we construct a splitting of the metaplectic cover of the reductive dual pairs of orthogonal and symplectic groups or the reductive dual pairs of unitary groups over a nonarchimedean local field with respect to a generalized lattice model of the Weil representation. We also prove a result concerning the splitting that we construct and the theta dichotomy for unitary group. The spl...

Characterizing the universal coverings of smooth projective varieties is an old and hard question. Central to the subject is a conjecture of Shafarevich according to which the universal cover X̃ of a smooth projective variety is holomorphically convex, meaning that for every infinite sequence of points without limit points in X̃ there exists a holomorphic function unbounded on this sequence. In t...

‎We first give bounds for domains where the unitarizabile subquotients can show up in the parabolically induced representations of classical $p$-adic groups‎. ‎Roughly‎, ‎they can show up only if the‎ ‎central character of the inducing irreducible cuspidal representation is dominated by the‎ ‎square root of the modular character of the minimal parabolic subgroup‎. ‎For unitarizable subquotients...

Given a nilpotent orbit O of a real, reductive algebraic group, a necessary condition is given for the existence of a tempered representation π such that O occurs in the wave front cycle of π. The coefficients of the wave front cycle of a tempered representation are expressed in terms of volumes of precompact submanifolds of an affine space.

D. Prasad showed that the sign of a self-dual representation finite or p-adic reductive group is often detected by central element. We study extension his results to some more general situations and make observations about consequences results.

The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f1, f2 . . . , fn be homogeneous invariant functions. Then the polarizations of f1, f2 . . . , fn define the nullcone of k ≤ m copies of V if and only if every linear subspace L of the nullcone of V of dimension ≤ m is annhilated by a one-parameter subgroup (shortly a 1-PSG). This m...

Abstract The problem of finding generators the subalgebra invariants under action a group automorphisms finite-dimensional Lie algebra on its universal enveloping is reduced to homogeneous same acting symmetric tensor algebra. This process applied prove constructive Hilbert–Nagata Theorem (including degree bounds) for in nilpotent relatively free associative endowed with an induced by represent...

Suppose G is a real reductive Lie group, with maximal compact subgroup K. The representation theory of K is well-understood and well-behaved: K̂ is a countable discrete set consisting of finite-dimensional representations (δ, Eδ). If (π, V ) is a quasisimple irreducible representation of G, HarishChandra proved that each irreducible representation of K appears at most finitely often in V ; so th...