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On the double-affine Bruhat order: the $$\varepsilon =1$$ conjecture and classification of covers in ADE type. (English) Zbl 1414.05304
For any Kac-Moody group $$G$$, the authors prove that the Bruhat order on the semidirect product $$W_{\mathcal{T}} = \mathcal{T} \rtimes W$$ of the Weyl group $$W$$ and the Tits cone $$\mathcal{T}$$ for $$G$$ is strictly compatible with a $$\mathbb{Z}$$-valued length function. In the proof, the inversion sets play an important role. For a positive root $$\beta \in G$$ and for an integer $$n\in \mathbb{Z}$$, there is an associated reflection element $$s_{\beta[n]}$$. In general $$s_{\beta[n]} \notin W_{\mathcal{T}}$$. For suitable $$x\in W_{\mathcal{T}}$$ with $$xs_{\beta[n]}\in W_{\mathcal{T}}$$, they construct an inversion set $$\mathrm{Inv}_{x}^{++}(s_{\beta[n]})$$ and prove that $$xs_{\beta[n]}>x$$ if and only if $$\mathrm{Inv}_{x}^{++}(s_{\beta[n]})$$. In particular, they prove that if $$xs_{\beta[n]}>x$$ then $$l(xs_{\beta[n]})-l(x)=\#(\mathrm{Inv}_{x}^{++}(s_{\beta[n]}))$$, where $$l:W_{\mathcal{T}}\rightarrow \mathbb{Z} \oplus \mathbb{Z}\varepsilon\rightarrow \mathbb{Z}$$ is the composed length function obtained by setting $$\varepsilon=1$$.
Then, they consider the question that for $$x,y\in W_{\mathcal{T}}$$ whether $$y$$ covers $$x$$ if and only if $$x<y$$ and $$l(x)=l(y)-1$$. They answer this question positively when $$G$$ is untwisted affine ADE type. For this type, using explicit control of the pairings between roots and coroots, they reduce the problem to a calculation that is essentially the case of affine $$\mathrm{SL}_2$$, where they verify this situation by explicit computations.

##### MSC:
 5e+10 Combinatorial aspects of representation theory
##### Keywords:
Kac-Moody groups; double-affine Bruhat order
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##### References:
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