Gail Letzter

Ring theory from symplectic geometry

Abstract Basic results for an algebraic treatment of commutative and noncommutative Poisson algebras are described. Symplectic algebras are examined from a ring-theoretic point of view.

SUBALGEBRAS WHICH APPEAR IN QUANTUM IWASAWA DECOMPOSITIONS

Let g be a semisimple Lie algebra. Quantum analogs of the enveloping algebra of the fixed Lie subalgebra are introduced for involutions corresponding to the negative of a diagram automorphism. These subalgebras of the quantized enveloping algebra specialize to their classical counterparts. They a re used to form an Iwasawa type decompostition and begin a study of quantum Harish-Chandra modules.

Harish-Chandra modules for quantum symmetric pairs

Let U denote the quantized enveloping algebra associated to a semisimple Lie algebra. This paper studies Harish-Chandra modules for the recently constructed quantum symmetric pairs U ,B in the maximally split case. Finite-dimensional U -modules are shown to be Harish-Chandra as well as the B-unitary socle of an arbitrary module. A classification of finite-dimensional spherical modules analogous to the classical case is obtained. A one-to-one correspondence between a large class of natural finite-dimensional simple Bmodules and their classical counterparts is established up to the action of almost B-invariant elements. Let g be a semisimple Lie algebra and let g be the Lie subalgebra fixed by an involution θ of g. There is an extensive theory concerning the Harish-Chandra modules associated to the pair g, g. These are g-modules which behave nicely with respect to the restriction of the action of g to g. One of the main motivations in understanding such modules is their close connection and impact on the study of real Lie group representations. Harish-Chandra modules also are a class of infinitedimensional g-modules that have a reasonable amount of structure and thus are amenable to study. In this way they are similar to the other infinite-dimensional g-modules that have been examined thoroughly, the so-called category O modules. These modules behave nicely with respect to the Cartan subalgebra of g. Let U denote the Drinfeld-Jimbo quantization of the enveloping algebra of g. In the quantum case, there is already a well developed theory of category O modules (see for example [Jo]). However, much less is known about infinite-dimensional quantum modules that correspond to the classical Harish-Chandra modules. One of the main reasons for this difference is that there is an obvious quantum analog of the Cartan subalgebra of g, while the analogs of the invariant Lie subalgebra are less apparent. In [L2], we introduced one-sided coideal algebrasB = Bθ as quantum analogs of the enveloping algebra of the fixed Lie subalgebra under the maximally split form of an involution θ. These analogs generalize the known examples already in the literature in the maximally split case. Using these analogs in this paper, we lay the groundwork for the study of quantum Harish-Chandra modules. In the first part of the paper, we prove elementary results about quantum Harish-Chandra modules associated to U,B. As in the classical case, every finitedimensional simple U -module comes equipped with a positive definite conjugate linear form. One checks that B behaves nicely with respect to this form which allows us to decompose finite-dimensional U -modules into a direct sum of finite-dimenReceived by the editors October 22, 1999 and, in revised form, November 19, 1999. 2000 Mathematics Subject Classification. Primary 17B37. The author was supported by NSF grant no. DMS-9753211. c ©2000 American Mathematical Society

On the Brylinski-Kostant filtration

The base field k is assumed to be of characteristic zero. Let g be a split semisimple k-Lie algebra. Consider a finite-dimensional simple g module V and fix a weight μ of V . This paper concerns the Brylinski-Kostant (or simply, BK) filtration defined on the μ weight space of V . In particular, the members of the n subspace in the filtration are those vectors of weight μ killed by the n power of a fixed regular nilpotent element. The q character corresponding to this filtration, referred to in [B1] as the jump polynomial, is the associated (finite) Poincaré series for the filtration in the variable q. A second q polynomial was introduced by Lusztig ([L1]); it is the coefficient of e in a q version of the ordinary character formula for V defined using a q analog of Kostant’s partition function (see Section 2.3 for a precise definition). The aim of this paper is to give a new proof of [B1, Theorem 3.4]: the jump polynomial of a dominant weight μ is equal to Lusztig’s q polynomial at μ. We also obtain a natural extension of this result to non-dominant weights. We briefly review some high points in the history of the BK filtration and its related jump polynomial. First, A. Shapiro and R. Steinberg independently found an empirical method of reading off the exponents in the Poincaré polynomial of the adjoint group from the root system for g. B. Kostant [K1] found the theoretical underpinnings of this procedure by studying the decomposition of g into submodules for the action of the principal TDS. This computes the BK filtration for the adjoint representation. In a later paper [K2], Kostant considered generalized exponents associated to any V as described above. As a consequence it is possible to obtain a remarkable relation between the “harmonic degrees” of V and what we now call the BK filtration of the zero weight space V0 of V , by combining [K2, Sect. 5, Cor. 4] and [B1, Lemma 2.5 and Prop. 2.6]. Central to the derivation of this observation is a difficult result of [K2] describing the ideal of definition for the nilpotent cone. Hesselink [H] and Peterson [P] then (independently) gave a purely combinatorial formula for these generalized exponents. Specifically, one can read the generalized exponents as Lusztig’s q polynomial at weight zero, thus establishing the first connection between Lusztig’s q formulas and jump polynomials. This combinatorial approach is useful, for example, in computing the so-called PRV determinants [J1].

Invariant differential operators for quantum symmetric spaces

This is the first paper in a series of two which proves a version of a theorem of Harish-Chandra for quantum symmetric spaces in the maximally split case: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Here, we establish this result for all quantum symmetric spaces defined using irreducible symmetric pairs not of type EIII, EIV, EVII, or EIX. A quantum version of a related theorem due to Helgason is also obtained: The image of the center under this Harish-Chandra map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not one of these four types.

Macdonald difference operators and Harish-Chandra series

We analyse the centralizer of the Macdonald difference operator in an appropriate algebra of Weyl group invariant difference operators. We show that it coincides with Cheredniks commuting algebra of difference operators via an analog of the Harish-Chandra isomorphism. Analogs of Harish-Chandra series are defined and realized as solutions to the system of basic hypergeometric difference equations associated to the centralizer algebra. These Harish-Chandra series are then related to both Macdonald polynomials and Chalykhs Baker-Akhiezer functions.

The center of quantum symmetric pair coideal subalgebras

The theory of quantum symmetric pairs as developed by the second author is based on coideal subalgebras of the quantized universal enveloping algebra for a semisimple Lie algebra. This paper investigates the center of these coideal subalgebras, proving that the center is a polynomial ring. A basis of the center is given in terms of a submonoid of the dominant integral weights.