Spaltenstein varieties of pure dimension
Yiqiang Li

TL;DR
This paper proves that Spaltenstein varieties for classical groups are pure dimensional under certain conditions and are Lagrangian in related geometric structures, extending results to $\sigma$-quiver-varieties.
Contribution
It establishes pure dimensionality and Lagrangian properties of Spaltenstein varieties for classical groups, extending to $\sigma$-quiver-varieties.
Findings
Spaltenstein varieties are pure dimensional for even or odd partitions.
They are Lagrangian in partial resolutions of nilpotent Slodowy slices.
Results extend to the $\sigma$-quiver-variety setting.
Abstract
We show that Spaltenstein varieties of classical groups are pure dimensional when the Jordan type of the nilpotent element involved is an even or odd partition. We further show that they are Lagrangian in the partial resolutions of the associated nilpotent Slodowy slices, from which their dimensions are known to be one half of the dimension of the partial resolution minus the dimension of the nilpotent orbit. The results are then extended to the -quiver-variety setting.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Algebra and Geometry · Algebraic Geometry and Number Theory
Spaltenstein varieties of pure dimension
Yiqiang Li
Department of Mathematics
University at Buffalo
the State University of New York
Abstract.
We show that Spaltenstein varieties of classical groups are pure dimensional when the Jordan type of the nilpotent element involved is an even or odd partition. We further show that they are Lagrangian in the partial resolutions of the associated nilpotent Slodowy slices, from which their dimensions are known to be one half of the dimension of the partial resolution minus the dimension of the nilpotent orbit. The results are then extended to the -quiver-variety setting.
Key words and phrases:
Spaltenstein varieties, pure dimensionality, nilpotent Slodowy slices of classical groups, symplectic geometry, -action, -quiver varieties.
2010 Mathematics Subject Classification:
14L35, 20G07, 51N30. 53D05
In memory of my uncle Renyi Huang
1. Introduction
1.1. Spaltenstein varieties
Let be a complex reductive group. Fix a parabolic subgroup of and a nilpotent element in Lie. The Spaltenstein variety of the triple is defined to be
[TABLE]
where is the nilpotent radical of . When is a Borel subgroup, a Spaltenstein variety is more commonly referred to as a Springer fiber [Spr76]. In general, a Spaltenstein variety is neither smooth nor irreducible. So an immediate question of substantial interest is if it is pure dimensional, that is if the dimensions of irreducible components of an are the same. It was answered in the affirmative in the following two fundamental cases by Spaltenstein [Sp76, Sp77] in the 70s, and independently by Steinberg [St74] for Case (a) when is a general linear group.
- (a)
is a Borel subgroup.
- (b)
is a general linear group.
Spaltenstein further provided an example in [Sp82, 11.6] showing that the variety is not always pure dimensional for a nilpotent element in of Jordan type . This example is recalled in Section 4, together with a few more in loc. cit. where can be described explicitly with fresh light casted upon. Beyond Steinberg and Spaltenstein’s results, little is known on the pure dimensionality of .
In this paper, we shall prove
Theorem A**.**
The Spaltenstein variety is pure dimensional if
- (c)
* is classical and the Jordan type of is an even or odd partition, i.e., of the form or .*
Our approach is to study Spaltenstein varieties in the context of symplectic geometry.
1.2. Symplectic geometry and -action
As is gradually known, complex symplectic geometry provides a new and conceptual way to understand the pure dimensionality of a complex variety. Precisely, there is the following remarkable result, whose proof can be found in the proof of Proposition 5.4.7 in [G09]. Note that we work in the setting of complex algebraic geometry.
Theorem B**.**
Suppose that is a proper morphism from a smooth symplectic algebraic variety, with algebraic symplectic -form, to an affine variety. Suppose further that both varieties admit a -action, compatible with . If the following two conditions hold:
- •
the -action provides a contraction of to its fixed-point locus ,
- •
the -action on has weight on the symplectic form on : , ,
then the fiber , or rather its associated reduced scheme, is Lagrangian in .
Being Lagrangian implies that is pure dimensional, provided that is so, and moreover its dimension is one half of that of .
It is exactly the framework of Theorem B that Spaltenstein variety is put under and that the proof of Theorem A falls out, which we shall discuss in more details as follows.
1.3. Slodowy slices and their partial resolutions
Retaining the setting in Section 1.1, the cotangent bundle of yields a partial resolution of singularities of the closure of a nilpotent orbit in for a Richardson element :
[TABLE]
Here the terminology ‘partial’ refers to the fact that the restriction of to the orbit is generically finite, but not isomorphic, in general. When is a Borel, the morphism is the Springer resolution to the nilcone of and a genuine resolution of singularities. On the other hand, fixing an -triple in , one can consider the Slodowy slice (see [Sl80]). Setting and (so that is nonempty if and only if ). The above map restricts to a partial resolution of the nilpotent Slodowy slice :
[TABLE]
Again when is a Borel, the morphism is a genuine resolution of singularities. The cotangent bundle carries a canonical symplectic structure, i.e., a closed -form, and from which the variety inherits one, say , as well. The variety is clearly an affine variety. Thanks to [BM83, Corollary 3.5 b)], it is know that
[TABLE]
In the cases (a) and (b) in Section 1.1, the above inequality becomes equality and is Lagrangian in . We shall show that the same holds for the case (c) in Theorem A. Moreover, is pure dimensional in general: it is a reduced complete intersection in of dimension (see [G08, Corollary 1.3.8]). Therefore we actually have a stronger version of Theorem A.
Theorem C**.**
If is a classical group and the Jordan type of is an even or odd partition, then the Spaltenstein variety is Lagrangian in in (9), and hence of pure dimension .
With the above discussion, the proof of Theorem C (and hence Theorem A) finally boils down to a search of the desired -actions for and to apply Theorem B.
Both varieties and admit a natural -action induced from the -triple so that is -equivariant. Moreover the -action provides a contraction of to , its -fixed point ([G08, 1.4]). However, the -action on the symplectic structure has weight , instead of weight as required in Theorem B. This defect is expected in light of Spaltenstein’s example: there is no uniform -action on and for all and satisfying all conditions in Theorem B.
Instead we obtain the desired -actions in the setting of Nakajima quiver varieties [N94, N98] and their variants in [Li19], from which this paper is grown out.
1.4. -action on Nakajima varieties
Thanks to the works of Nakajima [N94] and Maffei [M05], the proper map for being a general linear group has an incarnation as Nakajima quiver varieties attached to a type- quiver.
[TABLE]
Here and are tuples of integers determined by the Jordan types of the Richardson element and respectively, and is a generic parameter used for the stability condition. The orientation induces intrinsically a -action on the quiver varieties and . This action satisfies all conditions in Theorem B and hence provides a conceptual proof of the pure dimensionality of for being a general linear group, i.e., Case 1.1(b).
If is classical, i.e., an orthogonal or symplectic group, the map admits a quiver description , as a restriction of , in the recent work [Li19], with and realized as the fixed-point loci (resp. ) of Nakajima varieties (resp. ) under a specific involution :
[TABLE]
The -actions on Nakajima varieties can not be compatible with the involution in general, again due to Spaltenstein’s example. The crucial observation is that the place where the -action and the involution is compatible is where is Lagrangian. To this end, we show that the tuple under the conditions in Theorem C are the compatible places for the -action and the involution, hence providing a proof of Theorems C and A finally.
The arguments are indeed not restricted to type- graphs. We are able to establish a result that is valid for all Dynkin graphs. We drop the subscript in (3) and (4) to denote the morphism between Nakajima varieties of a fixed Dynkin graph.
Theorem D**.**
Assume that if there is an edge joining and . Then the fiber of the -fixed point under is Lagrangian in .
The main content of the paper is to study the compatibility of the -action and the automorphism in order to prove Theorem D. When the signature of the diagram isomorphism in the automorphism is , we can drop the assumption on in Theorem D and this more general result is stated in Theorem E.
1.5. Layout of the paper
In Section 2, we recall Nakajima varieties and their variants. In Section 3, we study the compatibility of -action with the various isomorphisms in the definition of -quiver varieties. In Section 4, we reproduce Spaltenstein’s examples in [Sp82, 11.6, 11.8] with new observations on being Lagrangian.
1.6. Acknowledgements
We thank the anonymous referee for helpful comments and insightful suggestions. This work was partly supported by the NSF grant DMS 1801915.
1.7. Updates after publication
(1). Theorem D implies the following result stronger than Theorem A: the variety is pure dimensional if
- (c’)
is classical and the Jordan type, say , of satisfies for all .
I thank Elek Balazs for pointing this out to me.
(2). A nilpotent orbit whose partition is either purely even or purely odd is called an even orbit in literature. I thank Bingyong Sun for pointing this out to me.
(3). The variety is invariant under row reduction and is conjectured to be true under column reduction. See the arXiv paper arXiv:2002.04422, Remark 2.4.3.
2. Preliminaries on quiver varieties
In the section, we recall briefly Nakajima varieties [N94, N98] and their variants in [Li19]. Our treatment follows closely Sections 1-4 in [Li19].
2.1. Nakajima varieties
Let be a Dynkin graph. Let and be the vertex and arrow set, respectively. For each arrow , let and be its outgoing and incoming vertex. There is an involution on the arrow set , such that and . Let and be two finite dimensional -graded vector spaces over the complex field of dimension vectors and , respectively. The framed representation space of the graph in is
[TABLE]
When and shall be highlighted, we write for . An element in is denoted by where is in , in and in . Let be an orientation function such that , . To a point , we set
[TABLE]
The space admits a symplectic structure with respect to given by
[TABLE]
Let act on from the left as follows. For all and , we define where , and for all and . Let
[TABLE]
be the moment map associated to the -action on the symplectic space . After identifying with its dual via the trace form, the -th component of is given by .
Let denote the -orbit of in .
Fix an embedding by for all . Let be the fiber . The group acts on .
Let . Fix an element in the first component of and an -graded subspace of , we say that is -invariant if for all . A point in is called -semistable if the following two stability conditions are satisfied. For any -graded subspaces and of of dimension and , respectively,
[TABLE]
Let be the -invariant set of all -semistable points in .
Let is the Cartan matrix of the graph . We set
[TABLE]
A parameter is called generic if it satisfies or . From now on, we assume that is generic. When is generic, the group acts freely on . Following Nakajima [N94, N98], we define the quiver variety attached to the data to be
[TABLE]
Let be the affinization of , with which is equipped a projective morphism Let be the image of under so that factors through a proper map under the same notation, which is (3) in type :
[TABLE]
The variety is smooth and symplectic with the latter induced from .
2.2. -quiver varieties
In this section, we recall -quiver varieties from [Li19].
2.2.1. Reflection functors
Recall the Cartan matrix . For each , we define a bijection by where , , . Let be the Weyl group generated by for all .
Let denote the vector whose -component is if and whose -th component is .
The reflection functor of Nakajima, Lusztig and Maffei [L00, M02, N03] associated to the simple reflection is defined to be
[TABLE]
where the pair () satisfies the conditions (R1)-(R4) as follows. Let be a vector space of dimension such that if and .
[TABLE]
Since if , we can define the reflection when , by switching the roles of and . So if and is generic, the reflection functor is defined to be the composition of the ’s:
[TABLE]
where is a composition of ’s.
2.2.2. The transpose
To any linear transformation between two vector spaces, each equipped with a non-degenerate bilinear form and , we define its right adjoint by the rule
[TABLE]
There is an isomorphism defined by .
Assume that the -th components and of and are equipped with non-degenerate bilinear forms for all . We define an automorphism
[TABLE]
where , and for all and . This automorphism induces an isomorphism:
[TABLE]
2.2.3. Diagram isomorphism
Let be an automorphism of , i.e., there are automorphisms of vertex and arrow sets, both denoted by , such that , and for all . Assume that is compatible with the function in the following sense. There exists a constant such that
[TABLE]
Let be the -graded vector space whose -th component is . The dimension vector of is whose -entry is . Given any point , we define a point by
[TABLE]
It induces a diagram isomorphism on Nakajima varieties:
[TABLE]
2.2.4. -Quiver varieties
Consider
[TABLE]
where , and are in (11), (10) and (13), respectively. The -quiver variety is defined by
[TABLE]
The proper map restricts to a proper morphism which is (4) in type :
[TABLE]
has a symplectic structure inherited from that of and is an affine variety as a closed subvariety of .
For the rest of this section, we consider the Dynkin graph of type : . Set if is an arrow from to and . The automorphism is the identity automorphism. The Weyl group element is the longest Weyl group element. Let where all components in is . For any pair , we define a new pair where
[TABLE]
Now set Let be a parabolic subgroup of a classical group whose levi has size indexed by . In other words, the isotropic flag variety is the collection of all isotropic flags such that the dimension difference of the -th step flag and -th step flag is . Note that may be empty. Let be the associated Richardson element. Let
[TABLE]
We write in Section 1.3 as when the Jordan type of is . The following result is obtained in [Li19, Corollary 8.3.4].
Proposition 2.3**.**
- (1)
If is equipped with a symmetric (resp*.** skew-symmetric) form for even (resp**.** odd), then and .* 2. (2)
If forms on are skew-symmetric (resp*.** symmetric) for even (resp**.** odd), then and .*
3. -action and the automorphism
In this section we assume that is generic and . We study the compatibility of modified version of a -action in [N94, Section 5] with the automorphism . By using these analyses, we then provide proofs for Theorems A-D.
3.1. Compatibility
To an orientation of , not necessarily the same as in the definition of Nakajima varieties, we can define two -actions on in (5). The first one is given by where
[TABLE]
The second one is given by where
[TABLE]
It is clear that each -action induces a -action on in (8), in light of the assumption that , but the induced ones on coincide as follows so that we do not have to distinguish the two actions on .
Lemma 3.2**.**
We have for all .
Proof.
Let . Then as required. ∎
It is clear that the weight of the symplectic form on with respect to this -action is , i.e., . Since the graph is Dynkin, the -action provides a contraction from , and hence , to its -fixed point .
The following lemma is the compatibility of the transpose in Subsection 2.2.2 and the -action.
Lemma 3.3**.**
We have for all and .
Proof.
We write and . We have
[TABLE]
This shows that , and the Lemma follows readily by Lemma 3.2. ∎
Let be an automorphism of . We assume that the pair is compatible with signature , see (12). We have the following compatibility of the automorphism and the -action.
Lemma 3.4**.**
Let be a compatible pair with signature . Then , for all and .
Proof.
We write and . We have
[TABLE]
So . The Lemma follows. ∎
The following lemma is the compatibility of the reflection functor and the -action.
Lemma 3.5**.**
We have for all and .
Proof.
Let . It suffices to show that the pair satisfies the conditions (R1)-(R4) in the definition of reflection functors. Recall and from (6). There is
[TABLE]
Thus we must have
[TABLE]
Clearly, is surjective since is so and is injective since is so. Hence (R1) holds for the pair . Similarly, there is
[TABLE]
This shows that the pair satisfies (R2). The condition (R3) for is clearly followed from definition. The condition (R4) for can be proved in a similar way as that of (R2). The Lemma thus follows. ∎
By combining Lemmas 3.3, 3.4 and 3.5, we have the following proposition.
Proposition 3.6**.**
Let be a compatible pair with signature . Then we have
[TABLE]
From Proposition 3.6 and the above analysis, we have readily
Proposition 3.7**.**
- (1)
If , for all and for all , then the -action in (18) on induces a -action on such that the weight of the symplectic form on is with respect to this -action. 2. (2)
If , for all and , then the -action provides a contraction of to its fixed-point consisting of a single point .
The following proposition provides compatible cases sufficient to prove our theorems.
Proposition 3.8**.**
- (1)
If , then , and . 2. (2)
Assume and if and are joined by an edge. Let be a partition satisfying the following conditions.
- •
For all , we have .
- •
For all , we have and .
Then for all and .
Proof.
The first statement is obvious. Let . It is enough to show that . Let be the parity of , i.e., if and if . Let . Then we have the following computations.
[TABLE]
Since for all , the above computation shows that . The proof is thus finished. ∎
3.9. The proof of Theorems A, C and D
Since is a Dynkin graph, hence bipartite, so we can find a partition of such that the first condition in Proposition 3.8 holds. Now set to be the unique orientation such that the second condition in Proposition 3.8 is valid. Since , we see that the automorphism is compatible with the orientation . In this case, the results in Proposition 3.7 are true and so Theorem B is applicable and from which Theorem D follows.
In light of Proposition 2.3, Theorem C, and hence Theorem A, follows from Theorem D. Note that we must show that all parabolic subgroups, up to conjugations, appear in the setting of Proposition 2.3. But this is already observed in Maffei’s work [M05, Theorem 8].
The proof of Theorems A, C and D is finished.
3.10. A generalization of Theorem D
In Proposition 3.8, there is no assumption on when , which is not stated in Theorem D, and the above argument works in this more general case as well. Let us record this more general result here.
Theorem E**.**
Let be a compatible pair of signature . Then the fiber is Lagrangian in .
4. Spaltenstein’s examples
In this section, we discuss examples in [Sp82, 11.6, 11.8], except 11.8 c). We show that is Lagrangian in all these examples, except the counterexample in [Sp82, 11.6].
4.1. [Sp82, 11.6]
Let us fix a basis of . Let be the bilinear form defined by for all , so that the associated symmetric matrix is the anti-diagonal identity matrix. Let be the special orthogonal group of and be its Lie algebra. Let be an element of the form
[TABLE]
Then it is clear that is of Jordan type and is a nilpotent element in . Let be the isotropic flag variety of isotropic subspaces in such that and . Then the Spaltenstein variety of the triple ( is the subvariety of consisting of elements such that There is a partition of where
[TABLE]
One can check that and are irreducible of dimension and , respectively. Indeed, for a fixed flag in , the freedom of is , the Grassmannian of isotropic lines in . The dimension of is , hence the dimension of is 3. For a fixed flag in , there is a unique flag , i.e., . Thus the dimension of is .
So the irreducible components of are of dimension and the closure of in of dimension 2. Hence is not pure dimensional.
Let be a parabolic subgroup such that is the isotropic flag varieties of all flags such that . From [Sp82, 11.6], is irreducible and of dimension 3. Let be the Richardson element associated to . Then it can be shown that , hence is Lagrangian in . This example is not in the cases (a)-(c) in the introduction.
4.2. [Sp82, 11.8. a)]
If is of type (resp. ; ; ; ), , with a Borel, and is minimal, then is a union of projective lines in a configuration of type (resp. or , the last is only possible if ; ; ; ). The condition implies that and is minimal implies that . So dimension of is 2, and thus is Lagrangian in .
4.3. [Sp82, 11.8. b)]
Let , with of type and a maximal isotropic Grassmannian. Then is a disjoint union of two projective lines. By Theorem C, is Lagrangian in .
4.4. [Sp82, 11.8. d]
Let and is a partial flag variety obtained from the complete flag by dropping the -th step for all , is a nilpotent of type . Then is a union of projective lines subject to certain conditions. From Theorem C, is Lagrangian in .
4.5.
By the rectangular symmetry in [Li19], one can produce more examples from previous subsections. For example, the corresponding case in Section 4.1 for is , is chosen such that is isomorphic to the isotropic flag varieties of with , and is of Jordan type . Then is not pure dimensional.
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