Let $G$ be a reductive group over complex numbers. Fix some maximal torus $T$. Let $\Lambda^{+}$ be the monoid of dominant coweights. It is known that one has a Cartan decomposition $$G((z))=\coprod\limits_{\lambda \in \Lambda^{+}}G[[z]]z^{\lambda}G[[z]]$$.

Is that true that the following equality holds too?

$$G[z,z^{-1}]=\coprod\limits_{\lambda \in \Lambda^{+}}G[z]z^{\lambda}G[z]$$

  • $\begingroup$ It's true for $G = GL_n$, by the elementary divisor theorem (more generally, for any PID $A$ and any prime element $z \in A$ we have a similar expression for $GL_n(A[z^{-1}])$). It then follows for $SL_n$ as well. $\endgroup$ – fherzig Jun 14 '17 at 16:08
  • $\begingroup$ I think the answer is yes, see Proposition 1.2.4 of Ginzburg's "Perverse sheaves on a loop group and Langlands duality". $\endgroup$ – uncookedfalcon Jun 14 '17 at 17:49

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