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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:
05E10 Combinatorial aspects of representation theory
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