Parabolic Hecke subalgebras, induction, and restriction

Let $(S,m)$ be a finite Coxeter matrix, $W$ its presented group with length $\mathrm{\ell }$ and standard parabolic subgroup $W_J=⟨J⟩$ for $J\subseteq S$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups). Let $R$, $H$, and the standard basis $\{T_w:w\in W\}$ be those of Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy and The standard basis of the generic Hecke algebra and base change. Use the inherited parameters $v_j$ on the restricted Coxeter system $W_J$; distinct odd-edge components of $J$ may therefore carry the same parameter when they lie in one odd-edge component of $S$.

Parabolic Hecke subalgebra. Define the candidate $R$-submodule

$$H_J:=\underset{u\in W_J}{⨁}R{T}_u\subseteq H\mathrm{.}$$

The justifying theorem Parabolic freeness and bases of induced Hecke modules ↗ proves it is a subalgebra, is generated by $T_j$ for $j\in J$, and is the embedded image of the restricted-system Hecke algebra over $R$. The same theorem proves the left and right freeness assertions needed by the induced-module construction below.

For a left $H_J$-module $M$, define its parabolic induction to be the balanced tensor product

$${\mathrm{Ind}}_J^W(M):=H{\otimes }_{H_J}M,$$

formed from the right $H_J$-module $H$ and left $H_J$-module $M$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums, Universal property of the tensor product for balanced maps into abelian groups). Thus $(hk)\otimes m=h\otimes (km)$ for $h\in H$, $k\in H_J$, and $m\in M$. The left $H$-action is $a\cdot (h\otimes m)=(ah)\otimes m$; it respects balancing because $(ahk)\otimes m=(ah)\otimes (km)$, and its module laws follow from associativity and the unit of $H$.

For a left $H$-module $N$, define parabolic restriction ${\mathrm{Res}}_J^W(N)$ to be its underlying $R$-module with the $H_J$-action obtained from the inclusion $H_J\hookrightarrow H$ proved by the same justifier. These constructions use left modules throughout; the theorem below also records the opposite right-module decomposition of $H$ so the two coset conventions remain explicit.