We give a stratification of the GIT quotient of the Grassmannian G 2,n modulo the normaliser of a maximal torus of SL n (k) with respect to the ample generator of the Picard group of G 2,n . We also prove that the flag variety GL n (k)/B n can be obtained as a GIT quotient of GL n+1 (k)/B n+1 modulo a maximal torus of SL n+1 (k) for a suitable choice of an ample line bundle on GL n+1 (k)/B n+1 .Let k be an algebraically closed field. Consider the action of a maximal torus T of SL n (k) on the Grassmannian G r,n of r-dimensional vector subspaces of an n-dimensional vector space over k. Let N denote the normaliser of T in SL n (k). Let L r denote the ample generator of the Picard group of G r,n . Let W = N/T denote the Weyl group of SL n (k) with respect to T .In [7], it is shown that the semi-stable points of G r,n with respect to the T -linearised ample line bundle L r is same as the stable points if and only if r and n are co-prime.In this paper, we describe all the semi-stable points of G r,n with respect to L r . In this connection, we prove the following result: First, we introduce some notation needed for the statement of the theorem.Let h j be a Cartan subalgebra of sl j +1 , P(h j ) be the projective space and R j ⊆ h * j be the root system. Let V j be the open subset of P(h j ) defined byHere, the Weyl group of sl j +1 is S j +1 , and h j is the standard representation of S j +1 .With this notation, taking m = n−1 2 (for this notation, see Lemma 1.6) and t = [ n−3 2 ] we have the following.Theorem. N\\G ss 2,n (L 2 ) has a stratification t i=0 C i where C 0 = S m+1 \P(h m ), and C i = S i+m+1 \V i+m .

In this paper, for any simple, simply connected algebraic group G of type B, C or D and for any maximal parabolic subgroup P of G, we describe all minimal dimensional Schubert varieties in G/P admitting semistable points for the action of a maximal torus T with respect to an ample line bundle on G/P . We also describe, for any semi-simple simply connected algebraic group G and for any Borel subgroup B of G, all Coxeter elements τ for which the Schubert variety X(τ ) admits a semistable point for the action of the torus T with respect to a non-trivial line bundle on G/B.

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In this note, we prove a theorem on a new presentation for the algebra of the endomorphisms of the permutation representation (Yokonuma-Hecke algebra) of a simple Chevalley group with respect to a maximal unipotent subgroup. This presentation is given using certain nonstandard generators.

The aim of this paper is to begin a study of the cohomology modules H i (X(w), L λ ) for non-dominant weights λ on Schubert varieties X(w) in G/B. The aim is to setup a combinatorial dictionary for describing the cohomology modules and give criteria for their vanishing.Here L λ denotes the line bundle on X(w) corresponding to the 1-dimensional representation of B given by the character λ.