We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra.
The purpose of this note is to prove, as Lusztig stated, that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nil-elliptic elementnt istcl wherel is the rank of the associated finite type Lie algebra.
In our paper [KR] we began a systematic study of representations of the universal central extension[InlineEquation not available: see fulltext.] of the Lie algebra of differential operators on the circle.
In the present paper we study the remaing nontrivial case, that of a negative central charge-N.
In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number.