Compactification of the Symplectic Group via Generalized Symplectic Isomorphisms

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic zero. We have a left (G×G)-action on G defined as (g1, g2) ·x := g1xg −1 2 . A (G×G)-equivariant embedding G ↪→ X is said to be regular (cf. [BDP], [Br, §1.4]) if the following conditions are satisfied: (i) X is smooth and the complement X \G is a normal crossing divisor D1 ∪ · · · ∪Dn. (ii) Each Di is smooth. (iii) Every (G×G)-orbit closure in X is a certain intersection of D1, . . . , Dn. (iv) For every point x ∈ X, the normal space TxX/Tx(Gx) contains a dense orbit of the isotropy group Gx. If G ↪→ X is a (G × G)-equivariant regular compactification of G, then a sum ∑ aiDi of the boundary divisors is (G×G)-stable. Let G̃→ G be a finite covering. If the line bundle O( ∑ aiDi) has a (G̃ × G̃)-linearization, then the vector space H(X,O( ∑ aiDi)) of global sections of O( ∑ aiDi) becomes a (G̃ × G̃)-module. Kato [Ka] and Tchoudjem [T] described the decomposition of this (G̃× G̃)-module into irreducible (G̃× G̃)-modules. Kausz constructed a regular compactification KGLn of the general linear group GLn in [Kausz1]. In [Kausz2] he described the structure of the (GLn × GLn)modules of global sections of line bundles associated to boundary divisors. Although he dealt with only the very special regular compactification KGLn, a good thing is that his description of the (GLn × GLn)-modules is canonical. More precisely, he constructed a canonical isomorphism between the (GLn ×GLn)-modules of global sections of line bundles associated to boundary divisors on KGLn and the (GLn ×GLn)-modules of global sections of line bundles on a product of flag varieties. The fact that the decomposition is canonical is important when we apply the compactification of G to the study of the moduli of G-bundles. In fact, Kausz used the canonical decomposition of the above (GLn × GLn)-modules, and proved the factorization theorem ([Kausz3]) of generalized theta functions on the moduli stack of vector bundles on a curve. (The factorization theorem has also been obtained by Narasimhan-Ramadas [N-Rd] and Sun [S1], [S2].) The purpose of this paper is to establish an analogue of the Kausz’s results to the symplectic group. If V is a finite dimensional vector space, the general linear group GL(V ) is regarded as a moduli space of isomorphisms V → V . In [Kausz1], Kausz introduced a generalized isomorphism. The compactification KGL(V ) of GL(V ) is the moduli space of generalized isomorphisms from V to V . Now suppose that V is endowed with a non-degenerate alternate bilinear form. The symplectic group Sp(V ) is regarded as a moduli space of symplectic isomorphisms V → V . As a symplectic analogue, we introduce a generalized symplectic