Demazure character formula in arbitrary Kac-Moody setting. (English) Zbl 0635.14023

Kumar, Shrawan

Invent. Math. 89, 395-423 (1987).

Let \(\mathfrak g\) be an arbitrary (not necessarily symmetrizable) Kac-Moody Lie algebra with a Cartan subalgebra \(\mathfrak h\), associated group \(G\), Borel subgroup \(B\), maximal torus \(T\) corresponding to \(\mathfrak h\), and Weyl group \(W\). For any \(w\in W\) let \(X_ w\) denote the corresponding Schubert variety in \(G/B\). The author proves in this general setting many properties of Schubert varieties known to hold in the finite case. Among other things he proves that \(X_ w\) is a normal Cohen-Macaulay variety, constructs an explicit rational resolution of \(X_ w\) and proves the Demazure character formula. The method used in the paper seems to give a new proof even in the finite case. In particular it does no rely on any characteristic \(p\) approach.
As a consequence the author gets the Weyl-Kac character formula and the denominator formula for arbitrary Kac-Moody algebras.

Reviewer: Tadeusz Józefiak (East Lansing)

MSC:

14M15	Grassmannians, Schubert varieties, flag manifolds
17B65	Infinite-dimensional Lie (super)algebras
22E65	Infinite-dimensional Lie groups and their Lie algebras: general properties

Keywords:

Kac-Moody Lie algebra; Schubert variety; normal Cohen-Macaulay variety; Weyl-Kac character formula

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