Special issue to the memory of T.A. SpringerMinimal reduction type and the Kazhdan–Lusztig map☆
Introduction
Throughout this paper, let be a connected reductive group over with Lie algebra . Fix a Borel subgroup with a maximal torus . Let denote the Lie algebra of . Let be the corresponding Weyl group. Let be the set of conjugacy classes in . The set is canonically independent of the choice of .1
Let be the nilpotent cone in . Let be the set of -orbits on under the adjoint action. This is a partially ordered set such that if and only if . For , let be the dimension of the Springer fiber of any .
The main results of this article can be summarized by saying that the following two diagrams are identical: Here, is a map that we will define in this paper using affine Springer fibers; is the map defined by Kazhdan–Lusztig [7, §9.1] using the loop group. The maps and are defined by Lusztig [11], [12] using the geometry of itself and not the loop group.
Let (resp. ) be the field (resp. ring) of formal Laurent (resp. Taylor) series in one variable . For an affine scheme over , let be the formal loop space of defined as the ind-scheme with -points for any -algebra . Let be the formal arc space of defined as the scheme with -points . For more details, see [14, §1]. In particular, we have the loop group and the arc group . Let be the Iwahori subgroup which is the preimage of under the reduction map . Let be the affine Grassmannian of , and be the affine flag variety of . We also have the loop Lie algebra whose -points are , and .
Let be the subset of topologically nilpotent and generically regular semisimple elements. The notation will denote topologically nilpotent regular semisimple elements in the loop space of various spaces related to . In this introduction, the various are only defined as subsets of -points of the relevant ambient spaces; they will be viewed as the sets of -points of schemes starting in Section 4.
For , we recall from [7, §0] the definition of affine Springer fibers: These are closed sub-ind-schemes of and respectively. In this paper, we always equip and with the reduced ind-scheme structure.
The centralizer is the loop group of a maximal torus of , and it acts on and .
Recall from [7, Lemma 2] that maximal tori of up to -conjugacy are in canonical bijection with . For , a maximal torus of of type can be abstractly constructed as follows: if has order and is a primitive th root of unity, then is the fixed point subgroup of the Weil restriction under the action that is on and .
A loop torus in is the loop group associated to a maximal torus of , so that . Therefore the loop tori in up to -conjugacy are .
We will say is of type if its centralizer is of type under the above-mentioned bijection. Let be the set of elements of type .
Recall that is called elliptic if for any . Let denote the set of elliptic conjugacy classes.
Let be the parahoric subgroup of the loop torus . A concrete description of will be given in the beginning of Section 7. Let be the Lie algebra of . Let be the linear subspace of topologically nilpotent elements. The quotient can be identified with . Let be the subset of topologically nilpotent generically regular semisimple elements.
Let be the discriminant function. It is characterized as the unique -invariant regular function whose restriction to takes the form , where runs over all roots of with respect to .
For of type , let By [7, last line in p.130] and [2], .
For , let
Definition 1.6 An element is called shallow of type , if is of type , and .
For we denote the set of shallow elements of type by . Similarly, for a loop torus , let be the set of shallow elements in .
Let . Let be the natural map. We still use to denote . Let denote the subset of topologically nilpotent and generically regular semisimple points. Then is characterized by .
For , let be the image of . Let be the image of . Clearly, , and . We will show in Section 4 that is a scheme with reasonable properties (it is locally closed of finite presentation in ).
In [7, §9] the authors define a map as follows. For and a generic lifting with reduction , is a topologically nilpotent regular semisimple element of type . Then . Here generic lifting means for some open dense subset .
Let . Then we have a reduction map sending to , well-defined up to the adjoint action of .
For each nilpotent orbit , let be the preimage of under . Let be the preimage of under the natural projection . Then if , is a fibration with fibers isomorphic to the Springer fiber for .
Definition 1.10 For , we denote by the subset of consisting of nilpotent orbits such that . We define to be the set of minimal elements in under the partial order inherited from that of . We call elements in the minimal reduction types of .
Our first result identifies constructs a map using minimal reduction types, and we prove that is a section of this map.
Theorem 1.11 For and any , is a singleton depending only on . We denote the only nilpotent orbit in for by . This defines a map: The Kazhdan–Lusztig map is a section of . In particular, is injective and is surjective.
Remark 1.12 It is conjectured in [7, §9.13] that is injective. The map has been calculated by Spaltenstein for classical groups [16], [17] and in most cases for exceptional groups [18]. In the remaining cases he gave conjectural answers (see [15, p. 1163] which verifies two more cases in ). In particular, was known to be injective for of classical type and of type and . Our argument is independent of Spaltenstein’s, and implies that his predictions in [18] for the remaining cases are correct.
The next result identifies the pair of maps with the pair of maps defined by Lusztig which we now recall.
In [11] Lusztig defines a map as follows. Let be an element with minimal length within its conjugacy class . Lusztig shows in [11, Theorem 0.4(1)] that among the unipotent orbits that intersect , there is a unique minimal orbit , and is defined to be the nilpotent orbit corresponding to . He proves that depends only on the conjugacy class and not on the choice of a minimal length element. He proves that is surjective [11, Theorem 0.4(2)] and constructs in [12, Theorem 0.2] a canonical section For , he proves that the function given by achieves its minimum at a unique (the “most elliptic” conjugacy class in the fiber of ), which is defined to be .
Theorem 1.14 We have . .
The equality was conjectured by Lusztig in [12].
Before stating further results, we make the following conjectures on the minimal reduction types.
Conjecture 1.15 For any , is a singleton.
Conjecture 1.16 1 Let and . Let . Then the centralizer acts transitively on (which is non-empty by Theorem 1.11 and discrete by Lemma 7.7).
In this paper we prove the following partial results towards these conjectures.
Theorem 1.17 Let be a reductive group and . Assume contains an element in a parabolic subgroup such that for each , one of the following holds either is the Weyl group of a Levi subgroup (containing ) of type or ; or is the Coxeter class in .
Then Conjecture 1.15 holds for shallow elements of type .
In particular, Conjecture 1.15 holds for of types and .
Theorem 1.18 Let be a reductive group and for some . Assume contains an element in a parabolic subgroup such that for each , one of the following holds either is of classical type or ; or is the Coxeter class in .
Then Conjecture 1.16 holds for and .
In particular, Conjecture 1.16 holds for of classical types and .
The proofs for these theorems are different for classical groups and for exceptional groups.
We introduce certain subvarieties of the affine Grassmannian called the skeleta (for any ). These are fixed points of the neutral components of loop tori on . For classical groups and for , we are able to describe the skeleta explicitly (see Sections 8 Proofs for type, 9 Proofs for type), from which we deduce the theorems for these groups. It is an interesting question to give explicit descriptions for skeleta in other exceptional types.
To prove Theorem 1.11, Theorem 1.14 for exceptional types, we use the properties of certain subsets of the arc space of the Chevalley base, with Theorem 6.1 being the key observation. The proof of Theorem 6.1 uses a flatness theorem of Bouthier, Kazhdan and Varshavsky [3] for certain maps between arc spaces. Although the constructions of and are logically independent of Lusztig’s maps and , our proof that they are equal in the exceptional case relies on Lusztig’s explicit calculation of in [11].
After several reduction steps, we reduce to treat only elliptic conjugacy classes in in almost simple groups. The actual proofs start in Section 8, at the beginning of which we list what remains to be proved after the reductions.
In Section 11.7 we comment on a possible way to relate and directly.
Along the way we also prove some results that are of independent interest: in Theorem 2.3 we give a formula for the minimal length in an elliptic conjugacy class of in terms of root data; in Corollary 4.6 we show that the partial order on defined by Spaltenstein depends only on and not on the root system of ; in Corollary 11.1 we show that and respect partial orders on and .
Section snippets
A formula for
In this section we relate and the minimal length for elements in . This result will be used in the proof of Theorem 1.11.
Notations involving a Levi subgroup
Let be a Levi subgroup. It induces a map sending a nilpotent orbit of to the nilpotent orbit in containing . This map is not injective in general. This is not the Lusztig–Spaltenstein induction.
If is a Levi subgroup of containing , then the Weyl group is a parabolic subgroup of . This induces a map Again is not injective in general.
If , denote by the conjugacy class of in ; denote by or simply its conjugacy
Fp constructible subsets
For a scheme of finite type over and , let be its truncated arc space representing the functor . Then the arc space is the limit with natural projections .
A subset is called fp constructible (where fp stands for “finite presentation”) if there is and a Zariski constructible subset such that . Similarly there is the notion of fp open, fp closed, fp locally closed subsets of .
Subsets of
Recall the discriminant function in Section 1.5 factors through , which we still denote by . It induces a map of arc spaces , the latter is an infinite-dimensional affine space. By definition, . From this we see that is the set of -points of the open subscheme of .
Let be the subset in where the valuation of the discriminant function is , i.e., . Denote .
The Kazhdan–Lusztig map
For and , by [7, Proposition 8.2] we have . The following result is the technical crux for proving the main results for exceptional groups. It uses a theorem of Bouthier–Kazhdan–Varshavsky [3] on the flatness of certain maps between arc spaces.
Theorem 6.1 Let and . Then . Moreover, is irreducible and See Section 5.1 for the definition of the codimension of in .
Proof Let . First we claim
The skeleton
In this section we introduce a class of subvarieties of the affine Grassmannian called skeleton, which will be used to prove the main results for classical groups.
For each conjugacy class , we fix a loop torus of type under the bijection in Section 1.4.
Recall that is the parahoric subgroup of . Concretely, if we identify as the torus over given by the fixed points of on (where is the order of , is a primitive th root of unity,
What remains to be shown
By Lemma 3.5, to prove the theorems stated in Section 1, we only need to consider the case is almost simple, and only need to consider one isomorphism class of among each isogeny class. It remains to show the following:
- (i)
For all and elliptic non-Coxeter, (as defined in Lemma 5.7) is a singleton and is equal to . This implies Theorem 1.14(1) and the full statement of Theorem 1.11 by Lemma 5.9 and Corollary 6.3. (Note that the case is the Coxeter class is already proved
Skeleta in type
Let for . Let be an elliptic loop torus in . In this case the condition in Lemma 7.5 usually does not hold. However the failure of that condition is mild and can be explicitly analyzed.
The elliptic loop tori have the same description as in type : for quadratic extensions , where is a separable -algebra of degree . We will use the notations from Section 8.3.
We may identify the quadratic space with where is
Proofs for exceptional types
Recall is Lusztig’s map from [11].
Lemma 10.1 Let be almost simple and of exceptional type. Let be two distinct conjugacy classes such that . Assume is elliptic, then .
Proof We check the tables in [11] for the calculation of in exceptional types. In most cases, from the Carter notation for the conjugacy classes in we can see . For example, if is labeled and is labeled , then is the Coxeter class in a Weyl subgroup
Comments
Our main results have the following consequences on the partial order on .
Corollary 11.1 Lusztig’s map (or the map ) is order preserving for and . The Kazhdan–Lusztig map (or Lusztig’s map ) induces an isomorphism of posets .
Proof (1) If , letting , we have , hence . Therefore there is an element in which is . Since , we have . (2) If for and , then hence . By
Acknowledgments
At various stages of my career I have been inspired by Springer’s work on Springer representations, regular elements in the Weyl group and the geometry of symmetric spaces. It is my honor to dedicate this paper to the memory of T.A. Springer.
I have benefitted a lot from conversations with R. Bezrukavnikov, Xuhua He and G. Lusztig on the topics studied in this paper. I would also like to thank the referee for useful suggestions.
References (22)
Minimal length elements in some double cosets of coxeter groups
Adv. Math.
(2007)A class of irreducible representations of a weyl group
Nederl. Akad. Wetensch. Indag. Math.
(1979)- et al.
Twisted loop groups and their affine flag varieties, with an appendix by t. haines and rapoport
Adv. Math.
(2008) On the kazhdan-lusztig map for exceptional Lie algebras
Adv. Math.
(1990)- J. Adams, X. He, S. Nie, Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes....
The dimension of the fixed point set on affine flag manifolds
Math. Res. Lett.
(1996)- A. Bouthier, D. Kazhdan, Y. Varshavsky, Perverse sheaves on infinite-dimensional stakcs, and affine Springer theory,...
Solutions d’équations à coefficients dans un anneau hensélien
Ann. Sci. Ecole Norm. Sup. (4)
(1973)- et al.
Codimensions of root valuation strata
Pure Appl. Math. Q.
(2009) - et al.
Fixed point varieties on affine flag manifolds
Israel J. Math.
(1988)
Cited by (8)
KAZHDAN-LUSZTIG MAP AND LANGLANDS DUALITY
2024, arXivThe Hitchin Image in Type-D
2023, arXivMulticomponent KP type hierarchies and their reductions, associated to conjugacy classes of Weyl groups of classical Lie algebras
2023, Journal of Mathematical PhysicsOn Lusztig's map for spherical unipotent conjugacy classes in disconnected groups
2023, Bulletin of the London Mathematical Society
- ☆
Supported by the Packard Foundation and the Simons Foundation .