Generic Hecke algebras in the infinite

TL;DR

This work develops a cohesive framework linking infinite Steinberg crystallographic complex reflection groups to braid theory and Hecke deformations. By constructing Coxeter-like, minimal reflection presentations for both genuine and non-genuine families, it identifies the topological content of the new x-relations via configuration spaces and proves a braid theorem that equates Artin-type presentations with braid groups. This braid-theoretic foundation enables a systematic deformation to generic Hecke algebras, unifying affine GDAHA deformations with the broad class of Steinberg crystallographic groups and illuminating isomorphisms with DAHA/EHA manifestations in types $A$, $D$, and $E$. The results pave the way for a comprehensive GDAHA program across crystallographic complex reflection groups, including extensions to remaining families and sporadics, and raise natural questions about freeness and KZ-type connections in this broader complex-analytic setting.

Abstract

Aiming for a revival of the theory of crystallographic complex reflection groups, we compute (minimal) Coxeter-like reflection presentations for the infinite families of those non-genuine groups which satisfy Steinberg's fixed point theorem. These new presentations behave à la Coxeter, encoding many of the group's properties at a glance, and their signature feature -- named the $x$-relation -- is fully understood in terms of configuration spaces. Crucially, the presentations further achieve the braid theorem, allowing to deform into the generic Hecke algebra. In particular, we revisit the affine GDAHA family in deformation terms of the most general class of Steinberg crystallographic complex reflection groups.

Figures (8)

• Figure 1: The $n=2$ visualization of the "elliptic braid relation" $[u_i,t_1^{-1}u_jt_1]=0,\ i<j$, under counterclockwise convention. E.g., the left pair of strands is read downwards exactly as $u_it_1^{-1}u_jt_1$. The equivalence of the two sides becomes manifest in the picture, even more so when seen from the top. In configuration terms, $t_i$ swaps strand $i$ over strand $i+1$ while $u_k$ loops the first strand counterclockwise around hole $k$.
• Figure 2: The $n=3$ visualization of the pushrelation $r_1t_2r_1=t_2t_1=r_2t_1r_2$, together with a meridional side view (bottom) simplifying $t_2t_1$ into the translation $\alpha e_1-\alpha e_3$ as expected by the special configuration condition. In configuration terms, $r_i$ swaps strand $i$ over strand $i+1$ while $t_j$, corresponding to the vector $\alpha e_j-\alpha e_{j+1}$, loops along the longitude strands $j$ and $j+1$ in opposite directions---with strand crossing ruled by the points' diagonal arrangement in the Conventions. E.g., $t_2$ loops the second point rightward which, from the meridional side view, is seen crossing over the third point but under the first.
• Figure 3: The $n=3$ visualization of the "special pushrelation" $r_0t_1=t_2^{-1}r_0^{-1}$, which is best seen from the top. In configuration terms, $r_0$ swaps the last strand over the first while looping them along the meridian (as translations now correspond to the first copy of the lattice) in opposite directions.
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Theorems & Definitions (38)

• Definition 1.1
• Theorem 1.2: PS2019
• Remark 1.3
• Remark 1.4
• Theorem 1.5: Malle1996Puente2017
• Theorem 1.6
• Remark 1.7
• Lemma 1.8
• Theorem 1.9
• Remark 1.10