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Let $G^\min$ be a minimal Kac-Moody group. There is an affine ind-variety structure on $G^\min$ such that multiplication induces a regular isomorphism of $U^- \times B^\min$ with an open subset $G^\min_0$ of $G^\min$ ([Ku], Prop. 7.4.11). This open subset is the set of elements of $G^\min$ which admit a ''Gaussian decomposition.''

I would like to understand the open set $G^\min_0$ explicitly in the affine type A case. The minimal Kac-Moody group in type $A_{n-1}^{(1)}$ is closely related to the group $\widehat{SL_n} := SL_n(\mathbb{C}[t,t^{-1}])$. (To be precise, a central extension of $\widehat{SL_n}$ is isomorphic to the quotient of $G^\min$ by its center ([Ku], Thm. 13.2.8 and Cor. 13.2.9).) Under this isomorphism, $B^\min$ is (modulo two dimensions in the torus) equal to the subgroup $$ \{M \in SL_n(\mathbb{C}[t]) : M(0) \text{ is upper triangular}\} $$ and $$ U^- = \{M \in SL_n(\mathbb{C}[t^{-1}]) : M(\infty) \text{ is lower triangular with 1's on the main diagonal}\}. $$

I have two questions:

  1. There is a natural ind-variety structure on $\widehat{SL_n}$, where the $n$th piece in the filtration consists of matrices whose entries are of the form $$a_{-n}t^{-n} + \ldots a_0 + \ldots a_n t^n.$$ Is this the same as the ind-variety structure on $G^\min$ coming from the double Bruhat cell decomposition and representation theory (as defined in [Ku], p.236)? If not, what ind-variety structure on $\widehat{SL_n}$ corresponds to that of $G^\min$?

  2. If the answer to the first question is yes, what are the defining equations of the open set $G^\min_0$? In type A, a matrix admits Gaussian decomposition if and only if the initial principal minors are nonzero; is there a similarly nice criterion in affine type A? For instance, elements of $\widehat{SL_n}$ can be viewed as infinite periodic matrices with finite support, so one could hope for a criterion involving certain minors of these matrices...

$\textbf{Reference}$:

([Ku]) S. Kumar, Kac-Moody groups, their flag varieties, and representation theory.

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