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Overview
The theory of D-modules deals with the algebraic aspects of differential equations. These are particularly interesting on homogeneous manifolds, since the infinitesimal action of a Lie algebra consists of differential operators. Hence, it is possible to attach geometric invariants, like the support and the characteristic variety, to representations of Lie groups. By considering D-modules on flag varieties, one obtains a simple classification of all irreducible admissible representations of reductive Lie groups. On the other hand, it is natural to study the representations realized by functions on pseudo-Riemannian symmetric spaces, i.e., spherical representations. The problem is then to describe the spherical representations among all irreducible ones, and to compute their multiplicities. This is the goal of this work, achieved fairly completely at least for the discrete series representations of reductive symmetric spaces. The book provides a general introduction to the theory of D-modules on flag varieties, and it describes spherical D-modules in terms of a cohomological formula. Using microlocalization of representations, the author derives a criterion for irreducibility. The relation between multiplicities and singularities is also discussed at length.
Originally published in 1990.
The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Product Details
| ISBN-13: | 9781400862078 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 07/14/2014 |
| Series: | Mathematical Notes , #1194 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 142 |
| File size: | 5 MB |
Read an Excerpt
D-modules and Spherical Representations
By Frédéric V. Bien
PRINCETON UNIVERSITY PRESS
Copyright © 1990 Princeton University PressAll rights reserved.
ISBN: 978-0-691-02517-9
CHAPTER 1
Localization Theory
I.1 Basic notions on D-modules
Let X be a smooth complex algebraic manifold. Let O = Ox be the sheaf of regular functions on X and D = Dx be the sheaf of differential operators on X. A D-module M is a sheaf on X which is quasi-coherent as am O-module and which has a structure of module over D. Let D-mod denote the category of left D-modules. We shall deal with some sheaves of rings slightly more general than D. Consider the category of pairs (A, iA) where A is a sheaf on X of C-algebras and iA : O -> A is a morphism of C-algebras. The pair (D, i) where i : O -> D is the natural inclusion will be called the standard pair.
1.1 Definition: A sheaf of twisted differential operators on X (tdo for short) is any pair (A, iA) which is locally isomorphic to the standard pair.
1.2 Lemma:The group of automorphisms of (i : O —> D) is naturally isomorphic to Z1(X) the group of closed 1-forms on X. The set of isomor- phism classes of tdo's on X is in bijection with the Cech cohomology space H1(X,Z1).
In particular tdo's form a linear space; for a proper variety X, H1(X, Z1) is the C-subspace of H2dR(X) generated by algebraic cycles.
1.3 Example: Let L be an invertible sheaf of O-modules on X, then the sheaf D(L) of differential operators acting on the local sections of L is a tdo. Let L correspond to the element c in the Picard group Hl{X, O*), then D(L) corresponds to the logarithmic differential of c, i.e. dc/c in H1(X, Z1).
1.4 Example: If X = Pn is the projective n-space, then H1(X, O*) = Z, hence, the invertible sheaves on Pn form a lattice. But H1(Pn, Z1) = C: the sheaves of twisted differential operators on Pn form a vector space.
1.5 Remark: If V is a locally free sheaf of O-modules on X of rank bigger than one, then the sheaf of rings of differential operators of V into itself is not a tdo. It consists of matrix differential operators. This sheaf will be useful in chapter IV.4.
Let f : X —> Y be a morphism of smooth complex algebraic varieties.
Let Oy, Ox be the sheaves of regular functions on Y and X. We will denote by fo, fa the inverse and direct images functors between the categories of O-modules. There is an induced map f* : Hl(Y, Z1) —> H1(X, Z1). Starting from a tdo (Dy,λ,i) on Y corresponding to [omega[ [member of] H1(Y, Z1), we can construct the tdo. (Dx,f•λ,f*i) on X which by definition is associated to the data f*ω [member of] H1(X, Z1). Let DY,λ-mod (resp. Dx,f•λ-mod) be the category of DY,λ-modules on Y (resp. Dx,f•λ-modules on X). Then one can define two functors :
• inverse image: f! : DY,λ-mod ->Dx,f•λ-mod
• direct image: f* : Dx,f•λ-mod —>DY,λ-mod
In case f is not affine, the direct image functor has good properties only between the derived categories of bounded complexes of D-modules. At the level of derived categories, the functor f! is simply the inverse image of O-modules shifted by the relative dimension of f : rd(f) = dim Y - dim X. One can transform a left D-module into a right one by tensorization with Ω the sheaf of top degree differential forms. Then the direct image of a right Dx,f•λ-module M. in the derived category is the direct image as O-module of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For the material of this section we refer to [Bernstein] and [Beilinson - Bernstein 1983].
1.2 D-modules with a group action
Let G be a complex algebraic group with Lie algebra g, and suppose G acts on a smooth variety X. Let F be a sheaf on X. By definition, a weak action of G on F is an action of G on F which extends the action of G on X. When F is a Dx-module, there is however more structure involved. If α denote the action of G on X, then we have a morphism dα : g -> Γ(X, DX), where Γ(X, F) denotes the global sections of Dx, i.e. the ring of differential operators defined on all of X. Γ(X, DX) acts on itself by the commutator action ad, so we get a map ad · da : g —> EndΓ(X, DX). On the other hand DX has obviously a weak action of G, say β, which yields the map dβ : g —> EndΓ(X, DX). It is natural to require that the map dα be G-equivariant and that ad · da coincide with dβ. More generally, let (D, i) be a sheaf of twisted differential operators on X.
2.1 Definition: An action of G on D is a weak action β together with a morphism π : g —> Γ(X, D) such that:
• π is G-equivariant with respect to Ad on g and β on Γ(X, D),
• for [xi] [member of] g : ad π([xi]) = dβ([xi]).
Suppose now that D is a tdo with a G-action given by β and π.
2.2 Definition: A (D, G)-module F on X is a quasi-coherent Ox-module with a structure γ of D-module and a weak action δ of G such that:
• γ is G-equivariant with respect to β on D and δ on F,
• for [xi] [member of] g : γπ([xi]) = dδ([xi])
The difference between a weak action and an action can also be illustrated as follows. Consider the diagram
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where a is the action morphism and p is the projection on the second factor. Then D has a G-action if and only if p°D [congruent to] a°D and a G-action on the D-module F is equivalent to the data of an isomorphism p°F [congruent to] a°F.
Decoding the definitions, one gets:
2.3 Proposition:Suppose that G acts transitively on X and let H be the stabilizer of a point x [member of] X.
1. The tdo on X with a G-action are in bijection with the tdo on {x} with an H-action, that is the 1-dimensional representations π ofh = Lie H.
2. The (D, G)-modules on X are in bijection with the (C, H)-modules on {x}, that is the representations (V, β) of H such that for [xi] [member of] h : π([xi]) = dβ([xi]).
2.4 Remark: From this proposition, we see that if H is connected, the sheaf V of sections of a G-homogenous vector bundle on X will not be a (D, G)-module when rank V > 1. In fact in this case, V is a module over a sheaf of matrix differential operators, as in Remark 1.5. We could enlarge the theory of D-modules to include the case of matrix differential operators acting on matrix valued functions. We will use these sheaves in chapter IV.4.
1.3 Parabolic subgroups and flag spaces
The structure theory of linear algebraic groups is clearly explained in Springer's book, so here we will only review certain facts from a point of view suited to the later developments.
The unipotent radical RuG of a linear algebraic group G is the largest closed, connected, unipotent normal subgroup of G. G is called reductive if RuG = {e}. Henceforth, G shall denote a complex, connected, reductive, linear algebraic group. The connectedness assumption is not at all necessary for the study of D-modules but it simplifies the exposition here.
A parabolic subgroup P of G is a closed subgroup such that the quotient variety G/P is complete. A Borel subgroup B is a connected and solvable subgroup of G which is maximal for these properties. One proves that a closed subgroup of G is parabolic if and only if it contains a Borel subgroup. Since G is connected, a parabolic subgroup is its own normalizer. We prefer to view the flag space X := G/P as the as the variety of all parabolic subgroups of G conjugate to P.
Let P be a parabolic subgroup of G and N = Np its unipotent radical. A Levi subgroup L of P is a closed subgroup such that the product map L × N -> P is an isomorphism of varieties. Then L is reductive, normalizes N and is the centralizer in G of a connected torus in P. We shall also say that L is a Levi subgroup of G.
If T is a maximal torus in P, there is a unique Levi subgroup of P containing T. All Levi subgroups of P are conjugate. The projection P —> P/N gives an isomorphism of any Levi subgroup of P with P/N. Therefore we call P/N the Levi factor of P and we denote it by Lp; it is canonically attached to P. The Levi factors Lp and Lp' of two conjugate parabolic subgroups P and P' are not necessarily canonically conjugate because we can twist a given isomorphism by an inner automorphism of Lp or Lp'.
Let P1 := (P, P) denote the commutator subgroup of P and put Tp = P/P1. Tp is a maximal torus, and up to covering, it is isomorphic to the center of Lp; we call it the Cartan factor of P. For two conjugate parabolic subgroups P and P', the Cartan factors Tp and Tp' are canonically conjugate. When P is a Borel subgroup, its Cartan factors coincides with its Levi factor and is isomorphic to a Cartan subgroup of G. This special case is good to bear in mind for understanding the sequel easily.
Let us denote by boldface letters the Lie algebras of the groups considered. To define the set of roots of tp in g, we use the following trick. Choose a Levi subgroup L of P and let C be the connected component of its center. Then C acts by the adjoint action on g and we obtain the roots of c in g. Note that these roots do not always form a root system. Now C is a finite cover of Tp by the map C [??] P —> Tp, hence to every root of c in g corresponds a unique linear form on tp. We call these linear forms: the roots of tp in g; they are independent of the choice of L.
Let R(tp) [subset] t*p be the set of roots of tp in g.R(tp) is naturally divided into the set of roots whose root spaces are contained in n and its complement. Let R+(tp) be the set of roots of tp in g/p. If α is a root of tp in g, the corresponding root space gα need not have dimension 1; dimgα is called the multiplicity of α. Let ρp be the half sum of the roots contained in R+(tp) counted with their multiplicities.
Let B be a Borel subgroup of G, contained in P. The map B/B1 —> P/P1 gives a canonical surjective homomorphism TB -> TP which dualizes to an inclusion t*p —> t*b. Hence we may think of R(tp) as a subset of t*b. R(tp) is not always a root system; for instance, let g = D4 and take a Levi factor whose semisimple part is of type A2. If we had chosen a invariant bilinear form on g*, then R(tp) could be viewed as the projection of R(tb) into t*p, and the multiplicity of α [member of] R(tp) would be the number of roots in R(tb) which project onto α.
We can also define the roots in the Levi factor lp as follows. Choose a maximal torus T in B. There is a unique Levi subgroup L of P containing T, so we have the roots of t in l. From the canonical isomorphism t*b [??] t*, we obtain the set R(tb, lp) of roots of tp in lp which is independent of the choice of T. R(tb, lp) [subset or equal to] R(tb) [subset] t*b. Set R+(tb, lp) = R+(tb) [intersection] R(tb, lp) and let ρl be the half sum of the roots in R+(tb, lp). We have ρl = ρb - ρp.
The outcome of this stylistic exercise is that we have a canonical comparison between the various root systems considered. To a Borel subgroup B of G, we can associate the triple (tb, R+(tp), R(tb)) where tb, = b/b1 and R+(tp) [subset] t*b. For different Borel subgroups, these triples are canonically conjugate. So we identify them with one abstract triple (t, R+, R) called the Cartan triple of G. The reductivity of G implies that R - R+ U - R+. Similarly to a pair B [subset or equal to] P we can associate the quadruple (t, tp, R+(t, lp), R+(tp)) where tp = p/p1, R+(tp) [subset] t*p [??] t*b [contains] R+(t, lp). Note that there is a canonical element ρl = ρb — ρp in t*.
Let D(t) be the set of simple roots in R+(t). The set D(lp) = D(t) [intersection] R(t, lp) of simple roots of L characterizes the conjugacy class of the parabolic subgroup P. Via this correspondence, the subsets of D(t) are in bijection with the G-conjugacy classes of parabolic subgroups of G.
If P is a parabolic subgroup of G corresponding to the subset I of D(t), let X := XI the space of all subgroups of G conjugate to P. X is called the flag space of G of type P or of type I; we will also use the words flag manifold or flag variety of type P. When P is a Borel subgroup of G, we call X the full flag variety of G. X is always a complete smooth complex projective algebraic variety on which G acts transitively by conjugation. If I [subset or equal to] J are two subsets of D(t), and if P is a parabolic subgroup of G of type I, there exists a unique parabolic subgroup Q of type J such that P [subset or equal to] Q. This yields a smooth fibration XI —> XJ with fiber isomorphic to the flag variety of Q of type P.
(Continues...)
Excerpted from D-modules and Spherical Representations by Frédéric V. Bien. Copyright © 1990 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
- FrontMatter, pg. i
- Acknowledgements, pg. vii
- Contents, pg. viii
- Introduction, pg. 1
- I. Localization Theory, pg. 9
- II. Spherical V-modules, pg. 29
- III. Microlocalization and Irreducibility, pg. 68
- IV. Singularities and Multiplicities, pg. 84
- Bibliography, pg. 125
- Index, pg. 130