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S. Kumar University of North Carolina Chapel Hill, United States shrawan@email.unc.edu

Descent of line bundles to GIT quotients of flag varieties
Let G be a connected semisimple complex algebraic group with a maximal torus T and let P be a parabolic subgroup containg T . We denote their Lie algebras by the corresponding Gothic characters. The following theorem is our main result. Theorem. Let L() be a homogeneous ample line bundle on the flag variety X = G/P . Then, the line bundle L() descends to a line bundle on the GIT quotient X ss ()//T (i.e., there exists a line bundle L on X ss ()//T whose pull-back to X ss () is the restriction of L()) if and only if for all the semisimple subalgebras s of g containing t (in particular, rank s = rank g), Z,

+

(s)

where + (s) is the set of positive roots of s. As a consequence of the above theorem, we get precisely which line bundles descend to the geometric quotients X ss ()//T . In the following Q (resp., ) is the root (resp., weight) lattice and we follow the indexing convention as in Bourbaki. Theorem. Let G be parabolic subgroup and X = G/P . Then, the X ss ()//T if and only if a connected, simply-connected simple algebraic group, P G a let L() be a homogeneous ample line bundle on the flag variety line bundle L() descends to a line bundle on the GIT quotient is of the following form depending upon the type of G.

a) G of type A ( 1) : Q b) G of type B ( 3) : 2Q c) G of type C ( 2) : Z21 + · · · + Z2
-1

+ Z

d1) G of type D4 : {n1 1 + 2n2 2 + n3 3 + n4 i : ni Z and n1 + n3 + n4 is even}. d2) G of type D ( 5) : {2n1 1 + 2n2 2 + · · · + 2n and n -1 + n is even }. e) G of type G2 : Z61 + Z22 . f ) G of type F4 : Z61 + Z62 + Z123 + Z124 . g) G of type E6 : 6P . h) G of type E7 : 12P i) G of type E8 : 60Q.
-2



-2

+n

-1



-1

+ n , ni Z

1