The cotangent bundle of a cominuscule Grassmannian
A theorem of the first author states that the cotangent bundle of the type Grassmannian variety can be embedded as an open subset of a smooth Schubert variety in a two-step affine partial flag variety. We extend this result to cotangent bundles of cominuscule generalized Grassmannians of arbitrary Lie type.
Earlier work of Lusztig and Strickland suggests possible connections between the conormal varieties to partial flag varieties on the one hand, and affine Schubert varieties on the other. In particular, Lusztig relates certain orbit closures arising from the type cyclic quiver to affine Schubert varieties [Lu90]. In the case , Strickland relates such orbit closures to conormal varieties of determinantal varieties [St82]; furthermore, any determinantal variety can be canonically realized as an open subset of a Schubert variety in the Grassmannian [LS78].
Inspired by these results, the first author was interested in finding a relationship between affine Schubert varieties and conormal varieties to the Grassmannian. As a first step, she showed that the compactification of the cotangent bundle to the Grassmannian is canonically isomorphic to a Schubert variety in a two-step affine partial flag variety [La14]. In this paper we extend her result to cominuscule generalized Grassmannians of arbitrary finite type (such Grassmannians occur in types ).
Let be a simple algebraic group over with associated Lie algebra and simple roots . A simple root is cominuscule if the coefficient of in any positive root of (written in the simple root basis) is less than or equal to .
The Weyl group of is generated by simple reflections corresponding to the the simple roots . For any subset , we let denote the parabolic subgroup whose Weyl group is generated by the elements of . For , set , so that is a maximal parabolic subgroup of . The manifold is called a generalized Grassmannian of type , and is said to be cominuscule if is cominuscule. For the remainder of the paper, we fix and consider the generalized Grassmannian associated to . Note that may or may not be cominuscule at this point.
Let denote the affine untwisted Kac-Moody algebra associated to , and let be the corresponding affine Kac-Moody group (see [Ku02, §6]).111We use calligraphic font (e.g. and ) for infinite-dimensional Kac-Moody groups, and non-calligraphic font for finite-dimensional Lie groups (e.g. , ). The Dynkin diagram for depends on the Dynkin diagram for , and is shown in Table LABEL:TBL:comin (see [Ca05, §18.1] or [Ka90, §4.8]). We use the convention that the affine node (sometimes called the special node) is labelled by zero, and similarly let and be the affine simple root and reflection respectively. The Weyl group of is generated by , and there is a parahoric subgroup associated to any subset . We let denote the associated affine flag variety, and denote the Weyl group of , or in other words the subgroup of generated by . For any subsets , let denote the set of minimal length coset representatives of . In particular, is the set of minimal length coset representatives of , and elements index Schubert varieties of .
Observe that . Let and . Let be the maximal element of , where . It is a standard fact that (see Lemma 2.3). The basis of this note is the following elementary but crucial observation:
If is cominuscule in then and are isomorphic.
The list of cominuscule simple roots in each type is well known. We indicate the cominuscule simple roots for each Dynkin diagram (up to diagram automorphism) in the left column of Table LABEL:TBL:comin, and the corresponding untwisted affine Dynkin diagram in the right column. In each case the Dynkin diagram of is isomorphic to the Dynkin diagram of , and this isomorphism identifies with the affine root . Consequently and are isomorphic. ∎
1.2. Results for cominuscule varieties
Consider the Schubert variety in . The Kac-Moody group acts on by left multiplication, and since is the Levi subgroup of , we can regard as a -variety.
In fact can naturally be considered as a -homogeneous fibre bundle over . More precisely:
The affine Schubert variety is stable under the left action of , and the natural projection is a -homogeneous fibre bundle map with fibre . In particular is smooth.
Our main result is that if is cominuscule then is a natural compactification of the cotangent bundle :
If is cominuscule, then the fibre is isomorphic to , and there is a -equivariant map of fibre bundles over , under which is isomorphic to a dense open subset of .
We prove Theorem 1.2 in Section 2 and Theorem 1.3 in Section 3. In order to prove Theorem 1.3 we explicitly construct the -equivariant embedding , which maps the base isomorphically onto the Schubert variety , and maps the fibre over the identity to a dense open subset of the Schubert variety .
When is minuscule rather than cominuscule, it is natural to replace with a twisted affine Kac-Moody group. Theorem 1.2 still holds in this case, but as we show in Section LABEL:S:minuscule, Theorem 1.3 does not hold. In this case the variety is not the compactification of the cotangent bundle , but of a different bundle over .
The second author thanks Calin Iuliu Lazaroiu and K. N. Raghavan for useful discussions. The Dynkin diagrams in tables LABEL:TBL:comin and LABEL:TBL:min are based on the excellent TikZ templates due to Oscar Castillo-Felisola. The first author was supported by NSA grant H98230-11-1-0197. The second author was partially supported by a fellowship from the Infosys Foundation.
2. The fibre bundle structure on
Given , we can write any uniquely as , where and . In this case the projection induces a projection , and the generic fibre of this projection is . We say is a parabolic decomposition with respect to .
For any , we define to be the set of simple reflections contained in a reduced expression for . For any , let , where is the Bruhat order on . We have the following proposition from [RS14, Theorem 2.3 and Proposition 3.2]:
The projection is a fibre bundle with fibre if and only if .
When the condition is satisfied, we say that is a Billey-Postnikov decomposition with respect to .
Recall that for any , we have if and only if is stable under left multiplication by the rank parahoric subgroup . It follows that if , then is stable under the action of the parahoric subgroup ([BL00], see also [RS14, Lemma 3.9]).
Let , so .
is a Billey-Postnikov decomposition with respect to .
Since is maximal in , we know that , for . It is clear that is a parabolic decomposition with respect to , and , proving part (a).
For part (b), if then , and hence , where and . Similarly , where and . So , and hence . But must be a proper subset of , so . ∎
Given , the Levi subgroup of is a Kac-Moody group with Weyl group . Since is affine, if is a strict subset of then is finite-dimensional, and similarly is finite. In order to prove Theorem 1.2 we need the following standard lemma:
If and , then is the parabolic subgroup of corresponding to the subgroup , and is isomorphic to a Schubert variety in the flag variety . In particular, if is the maximal element of then is isomorphic to .
Proof of Theorem 1.2.
By part (b) of Lemma 2.2, the variety is stable under the left action of . The base is clearly -stable as well, and the natural projection is -equivariant. By part (a) of Lemma 2.2 and Proposition 2.1, the projection is a -homogeneous fibre bundle with fibre .
Now the Levi subgroup of is simply . Since and is the maximal element of , we can alternately set and in Lemma 2.3 to get . Similarly is isomorphic to the flag variety . Since is a fibre bundle with smooth fibre and base, it follows that is smooth. ∎
3. The cotangent bundle
If is cominuscule then is isomorphic to by Lemma 1.1. To prove Theorem 1.3, we explicitly construct the map . Let be the Borel subgroup of the Kac-Moody group (in the literature is also known as the Iwahori subgroup of ). For convenience, we write for the Levi subgroup of the parahoric subgroup , where (in particular is the same as before). We let be the induced Borel of , and . Finally, let be the unipotent radical of . As in the previous section, , , and moreover for .
We will also need to use the underlying Lie algebras. We assume the standard construction of , in which
as a vector space (see [Ca05, §18.1] or [Ka90, §7.2]). Let be the Cartan subalgebra of the Kac-Moody algebra . Let be the (finite-dimensional) Lie algebra of , where . Let be the (nilpotent) Lie algebra of . Finally, let be the opposite nilpotent radical to inside . We consider the linear map
In order to prove Theorem 1.3 we will need the following lemma.
The map is a -equivariant isomorphism of vector spaces.
Let denote the set of roots of , with simple roots . The simple roots of and are the subsets of obtained by omitting and respectively. For any subalgebra , we let denote the set of -weights of , and let and denote the subsets of positive and negative roots respectively. Let be the highest root of , and let be the basic imaginary root of ([Ca05, §17.1] or [Ka90, §5.6]).
We can describe the set of roots of by
The set of positive roots of is given by
Note that and . Using the simple roots of and , the roots of and can then be written
where the requirement that (resp. ) follows from the fact that is cominuscule in (resp. is cominuscule in ).
Every root of can be written uniquely as where for all . Since is cominuscule, the coefficient of in is equal to . Using the previous description of , it follows that is an element of if and only if
for some coefficients (in particular, note that any of this form cannot belong to , since it will have positive -coefficient).
Note that for any , the homomorphism maps isomorphically onto . Thus the -weights of are precisely
This latter set is exactly the negative of the -weights of , since , and since is cominuscule in as in Lemma 1.1. We conclude that . Since is a clearly bijective, it is a vector space isomorphism.
Consider the left adjoint action of on . Under this action, each element of the weight space maps into whenever , and annihilates otherwise. Recall that , and observe that both and are stable under the left adjoint action of , and moreover that is -equivariant. It follows that is -equivariant. ∎
Using the map , we construct a map
is a -equivariant algebraic isomorphism from to an open dense subset of .
The exponential map is an algebraic isomorphism, and is an open dense subset of , where is the identity. Since is a -equivariant bijection and , the result follows. ∎
We can now finish the proof of the main theorem.
Proof of Theorem 1.3.
As in Section 2, we let , where . The cotangent bundle of is
the quotient of by the -action . We can define a map
where we use the fact that , which is stable under the left action of by Theorem 1.2. But is -equivariant, so we get an induced map
The cotangent bundle map sends . Since the projection sends , we conclude that the diagram