Torus quotients of Richardson varieties

Torus quotient of Richardson varieties in orthogonal and symplectic grassmannians

“…In this section, we provide a criterion for Richardson varieties in G/P r to admit semistable points with respect to L r . This criterion was proved for type A in [10]. Proposition 3.1.…”

“…For G is of type A in [10] it is shown that the above conditions are also sufficient. For type B, C and D the example below shows that the conditions w(χ) ≤ 0 and v(χ) ≥ 0 are only necessary but not sufficient.…”

“…A Richardson variety X v w in G/P is the intersection of the Schubert variety X w in G/P with the opposite Schubert variety X v therein. For G = SL n and P a maximal parabolic in G a criterion for the Richardson varieties in G/P to have nonempty semistable locus with respect to an ample line bundle L on G/P is given in [10]. In this paper, we give a criterion for a Richardson variety in G/P to have nonempty semistable locus with respect to the action of a maximal torus T on G/P , where G is of type B, C and D and P is a maximal parabolic subgroup in G.…”

Torus quotients of Richardson varieties in the Grassmannian

“…Later Bakshi-Kannan-Subrahmanyam [2] corrected the error and came up with a simpler proof for the same. In [9], Kannan-Paramasamy-Pattanayak-Upadhyay gave a condition on v and w for which the semistable locus in the Richardson variety X v w in Gr k,n is non empty. Bakshi-Kannan-Subhrahmanyam [2], showed that there is a unique minimal dimensional Richardson variety…”

“…, a k ). In [9], Kannan-Paramasamy-Pattanayak-Upadhyay gave a condition on v and w for which the semistable locus in the Richardson variety X v w in Gr k,n is non empty. The GIT quotient T \ \X(w k,n ) ss T (L) is shown to be smooth in [2] by Bakshi-Kannan-Subrahmanyam.…”