On the set of orbits for a Borel subgroup.

*(English)*Zbl 0828.22016Let \(X = G/H\) be a homogeneous variety for a connected complex reductive group \(G\) and let \(B\) be a Borel subgroup of \(G\). In many situations, it is necessary to study the \(B\)-orbits in \(X\). An equivalent setting of this problem is to analyze \(H\)-orbits in the flag variety \(G/B\). The probably best known example is the Bruhat decomposition of \(G/B\) where one takes \(H = B\). Another well studied situation is the case where \(H\) is a symmetric subgroup, i.e. the fixed point group of an involution of \(G\). Then \(H\)-orbits in \(G/B\) play a very important role in representation theory. They are the main ingredients for the classification of irreducible Harish-Chandra modules. In this paper, we introduce two structures on the set of all \(B\)-orbits. The first one is not really new, namely an action of a monoid \(W^*\) on the set \({\mathfrak B} (X)\) of all \(B\)-stable closed subvarieties of \(X\). As a set, \(W^*\) is the Weyl group \(W\) of \(G\) but with a different multiplication. That has already been done by R. Richardson and T. Springer [Geom. Dedicata 35, 389-436 (1990; Zbl 0704.20039)] in the case of symmetric varieties and the construction generalizes easily. As an application we obtain a short proof of a theorem of M. Brion [Manuscr. Math. 55, 191-198 (1986; Zbl 0604.14048)] and E. Vinberg [Funct. Anal. Appl. 20, 1-11 (1986; Zbl 0601.14038)]: If \(B\) has an open orbit in \(X\) then \(B\) has only finitely many orbits. Varieties with this property are called spherical. All examples mentioned above are of this type.

The second structure which we are introducing is an action of the Weyl group \(W\) on a certain subset of \({\mathfrak B} (X)\). Let me remark that in the most important case, \(X\) spherical, \({\mathfrak B} (X)\) is just the set of \(B\)-orbit closures and the \(W\)-action will be defined on all of it. We give two methods to construct this action. In the first, we define directly the action of the simple reflections \(s_\alpha\) of \(W\). This is done by reduction to the case \(\text{rk }G= 1\) and then by a case-by- case consideration. The advantage of this method is that it is very concrete and works in general. The problem is to show that the \(s_\alpha\)-actions actually define a \(W\)-action. For that the braid relations have to be verified which I don’t know how to do directly.

The second method does not have this problem but it is more complicated, less explicit, and works only in the spherical case. It is based on a construction of G. Lusztig and D. Vogan [Invent. Math. 71, 365-379 (1983; Zbl 0544.14035)]. Let \({\mathcal H}_q\) be the Hecke algebra attached to the Weyl group \(W\). Then Lusztig and Vogan define an \({\mathcal H}_q\)-module \({\mathcal C}_q\) which is closely related to \({\mathfrak B} (X)\).

In this paper we look only at the case \(q =1\). Then \({\mathcal H}_1\) is just the group algebra of \(W\), hence \({\mathcal C}_1\) is a \(W\)-module. We show that, after some modifications, \({\mathcal C}_1\) becomes a permutation representation with \({\mathfrak B} (X)\) as basis. Hence, this defines a \(W\)-action on \({\mathfrak B} (X)\). It should be noted that one could generalize Lusztig-Vogan’s construction of the \({\mathcal H}_q\)- module to all spherical varieties. Then specializing \(q = 0\) or \(q = \infty\) gives the \(W^*\)-action (see the paper by R. Richardson and T. Springer in [R. S. Elman (ed.) et al., Linear algebraic groups and their representations. Contemp. Math. 153, 109-142 (1993)] 7.4 in the symmetric case). Hence, \({\mathcal H}_q\) unifies both the \(W\)- and the \(W^*\)-action. However, in this paper I do not pursue this line any further. Actually, there is a third method to construct the \(W\)-action, but so far it works only on an even smaller subset of \({\mathfrak B} (X)\). It consists in relating \(B\)-orbits via conormal bundles to the cotangent bundle of \(X\). The advantage of this construction is that one obtains more information. Observe that \({\mathfrak B} (X)\) contains a distinguished element, namely \(X\) itself. We are able to determine its isotropy group \(W_{(X)} : W_{(X)} = W_X \ltimes W_{P(X)}\). Here \(W_X\) is the Weyl group of \(X\). It was defined by M. Brion [J. Algebra 134, 115-143 (1990; Zbl 0729.14038)] for spherical varieties and generalized by the author in [Invent. Math. 99, 1-23 (1990; Zbl 0726.20031); ibid. 116, 309-328 (1994; Zbl 0802.58024); Math. Ann. 295, 333-363 (1993; Zbl 0789.14040)]. The group \(W_{P(X)}\) is the Weyl group of a certain parabolic subgroup attached to \(X\). If \(X\) is symmetric \(W_X\) is just the little Weyl group and \(P(X)\) the complexification of a minimal parabolic subgroup. As opposed to the symmetric case, the definition of \(W_X\) is in general very complicated. Hence, it is one of the main virtues of the \(W\)-action on \({\mathfrak B} (X)\) that one obtains a relatively easy construction of \(W_X\). Finally, let me mention that many statements hold over an arbitrary algebraically closed ground field of characteristic \(p \geq 0\). The monoid action goes through and also the \(W\)-action, at least if \(X\) is spherical and \(p \neq 2\). There are counterexamples for \(p = 2\).

The second structure which we are introducing is an action of the Weyl group \(W\) on a certain subset of \({\mathfrak B} (X)\). Let me remark that in the most important case, \(X\) spherical, \({\mathfrak B} (X)\) is just the set of \(B\)-orbit closures and the \(W\)-action will be defined on all of it. We give two methods to construct this action. In the first, we define directly the action of the simple reflections \(s_\alpha\) of \(W\). This is done by reduction to the case \(\text{rk }G= 1\) and then by a case-by- case consideration. The advantage of this method is that it is very concrete and works in general. The problem is to show that the \(s_\alpha\)-actions actually define a \(W\)-action. For that the braid relations have to be verified which I don’t know how to do directly.

The second method does not have this problem but it is more complicated, less explicit, and works only in the spherical case. It is based on a construction of G. Lusztig and D. Vogan [Invent. Math. 71, 365-379 (1983; Zbl 0544.14035)]. Let \({\mathcal H}_q\) be the Hecke algebra attached to the Weyl group \(W\). Then Lusztig and Vogan define an \({\mathcal H}_q\)-module \({\mathcal C}_q\) which is closely related to \({\mathfrak B} (X)\).

In this paper we look only at the case \(q =1\). Then \({\mathcal H}_1\) is just the group algebra of \(W\), hence \({\mathcal C}_1\) is a \(W\)-module. We show that, after some modifications, \({\mathcal C}_1\) becomes a permutation representation with \({\mathfrak B} (X)\) as basis. Hence, this defines a \(W\)-action on \({\mathfrak B} (X)\). It should be noted that one could generalize Lusztig-Vogan’s construction of the \({\mathcal H}_q\)- module to all spherical varieties. Then specializing \(q = 0\) or \(q = \infty\) gives the \(W^*\)-action (see the paper by R. Richardson and T. Springer in [R. S. Elman (ed.) et al., Linear algebraic groups and their representations. Contemp. Math. 153, 109-142 (1993)] 7.4 in the symmetric case). Hence, \({\mathcal H}_q\) unifies both the \(W\)- and the \(W^*\)-action. However, in this paper I do not pursue this line any further. Actually, there is a third method to construct the \(W\)-action, but so far it works only on an even smaller subset of \({\mathfrak B} (X)\). It consists in relating \(B\)-orbits via conormal bundles to the cotangent bundle of \(X\). The advantage of this construction is that one obtains more information. Observe that \({\mathfrak B} (X)\) contains a distinguished element, namely \(X\) itself. We are able to determine its isotropy group \(W_{(X)} : W_{(X)} = W_X \ltimes W_{P(X)}\). Here \(W_X\) is the Weyl group of \(X\). It was defined by M. Brion [J. Algebra 134, 115-143 (1990; Zbl 0729.14038)] for spherical varieties and generalized by the author in [Invent. Math. 99, 1-23 (1990; Zbl 0726.20031); ibid. 116, 309-328 (1994; Zbl 0802.58024); Math. Ann. 295, 333-363 (1993; Zbl 0789.14040)]. The group \(W_{P(X)}\) is the Weyl group of a certain parabolic subgroup attached to \(X\). If \(X\) is symmetric \(W_X\) is just the little Weyl group and \(P(X)\) the complexification of a minimal parabolic subgroup. As opposed to the symmetric case, the definition of \(W_X\) is in general very complicated. Hence, it is one of the main virtues of the \(W\)-action on \({\mathfrak B} (X)\) that one obtains a relatively easy construction of \(W_X\). Finally, let me mention that many statements hold over an arbitrary algebraically closed ground field of characteristic \(p \geq 0\). The monoid action goes through and also the \(W\)-action, at least if \(X\) is spherical and \(p \neq 2\). There are counterexamples for \(p = 2\).

Reviewer: Reviewer (Berlin)

##### MSC:

22E46 | Semisimple Lie groups and their representations |

22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |

22E10 | General properties and structure of complex Lie groups |

20G15 | Linear algebraic groups over arbitrary fields |

20G05 | Representation theory for linear algebraic groups |