Spherical orbits and representations of Uε(g)

Abstract

Let Uε(g) be the simply connected quantized enveloping algebra at roots of one associated to a finite dimensional complex simple Lie algebra g. The De ConciniKac-Procesi conjecture on the dimension of the irreducible representations of Uε(g) is proved for the representations corresponding to the spherical conjugacy classes of the simply connected algebraic group G with Lie algebra g. We achieve this result by means of a new characterization of the spherical conjugacy classes of G in terms of elements of the Weyl group.

On the coordinate ring of spherical conjugacy classes

“…We proved that for every spherical conjugacy class O in G, there exists w ∈ W such that O ∩ X w = ∅ and ℓ(w) + rk(1 − w) = dim O: this then allows to prove the De ConciniKac-Procesi conjecture for simple U ε (g)-modules over elements in O. In fact we proved also a result in the opposite direction, giving therefore a characterization of spherical conjugacy classes in terms of the Weyl group ( [9], Theorem 25):…”

“…For length reasons we shall give proofs only for some classes. In [9] for the classical groups we gave representative of semisimple conjugacy classes in SL(n), Sp(n) and SO(n). Here we shall give an expression in terms of exp.…”

“…In [9] we exhibit the element x −β 1 (1) · · · x −β ℓ (1) ∈ O ∩ BwB ∩ B − . By Lemma 4.1, we can choose…”

“…In [9] we proved the De Concini-Kac-Procesi conjecture on the quantized enveloping algebra U ε (g) (introduced in [14]) for simple U ε (g)-modules over spherical conjugacy classes of G (we recall that a conjugacy class O in G is called spherical if a Borel subgroup of G has a dense orbit in O): our main tool was the representation theory of the quantized Borel subalgebra B ε introduced in [15].…”

“…Moreover w is always an involution (see [9], Remark 4, [10], Theorem 2.7). From this result we conjectured that, for a spherical O, the decomposition of the ring C[O] of regular functions on O (to which we refer as to the coordinate ring of O) as a G-module should be strictly related to w(O).…”