Parabolic induction, central orbit blocks and weight filtrations

Every right coset of $W_P$ has a minimal-length representative $w$; minimality gives $w(P)\subset R_L^{+}$, since otherwise right multiplication by a corresponding simple reflection decreases its inversion number. Such a $w$ sends all positive subsystem roots to positive roots. Along a reduced word for $v\in W_P$, inversion numbers therefore increase at each step of $wv$, giving $\mathrm{\ell }(wv)=\mathrm{\ell }(w)+\mathrm{\ell }(v)$. Conversely this equality shows $w$ is the unique minimum in its coset: if $u=wv$ were another minimum its length would be $\mathrm{\ell }(w)+\mathrm{\ell }(v)$, forcing $v=1$. Thus $W^P$ is exactly this set of representatives. Applying [F1] to finite products and the cross-relations shows $H_P$ has basis $N_v{\theta }_{\mathrm{\ell }}$ for $v\in W_P$, $\mathrm{\ell }\in L$: these elements span the subalgebra because cross-relations and finite Hecke multiplication keep all finite Weyl indices in $W_P$, and are independent as a subset of the PBW basis of $H$. The length-additive factorization $N_{wv}=N_w{N}_v$ now proves $H={⨁}_{w\in W^P}{N}_w{H}_P$. Tensoring gives $I_P(V)={⨁}_{w\in W^P}{N}_w\otimes V$, including the injected identity summand, which is $H_P$-stable.

Repeatedly move $f\in A$ through a reduced word for $N_w$ using [F1]. The leading term is $N_w{w}^{-1}(f)$; each correction omits at least one generator, so after finite Hecke multiplication it is a linear combination of $N_v{g}_v$ with $\mathrm{\ell }(v)<\mathrm{\ell }(w)$ and $g_v\in A$. Factoring each $v=up$ with $u\in W^P$, $p\in W_P$, gives $\mathrm{\ell }(u)\le \mathrm{\ell }(v)<\mathrm{\ell }(w)$. The subspaces of $I_P(V)$ spanned by summands with representative length at most $k$ are therefore $A$-stable. Their successive quotients are direct sums indexed by the representatives of length $k$, with $f$ acting on the $w$-summand as $w^{-1}(f)$ on $V$. Commuting invertible operators ${\theta }_{\mathrm{\ell }}$ can be simultaneously decomposed into generalized eigenspaces: take one lattice basis, split into generalized eigenspaces of the first operator, and successively split each stable summand for the remaining finitely many operators. In an invariant-subspace exact sequence the generalized $t$-weight spaces form an exact sequence, since their projections are polynomials in these commuting operators (finite-dimensional polynomial interpolation). This proves the multiset formula, including multiplicities and exactness under submodules and quotients.

For $f\in A^{W_P}$ every simple cross-relation in $P$ has zero right side, so $f$ commutes with $A$ and the finite generators, and belongs to the center of $H_P$. Distinct finite $W_P$-orbits in $T$ are separated by such invariants. To see this explicitly for two orbits, on their finite union choose for each ordered pair of distinct points a monomial taking different values, and multiply normalized linear differences to obtain a Laurent polynomial equal to one at a chosen point and zero at all others. Sum these polynomials over the second orbit and average over $W_P$; the resulting invariant is zero on the first orbit and one on the second. On an irreducible finite-dimensional complex module a central operator is scalar: any one of its eigenspaces is a nonzero stable subspace and hence the whole module. The invariant separation just proved therefore forces all its weights into one orbit. Now let $U$ be generated by a generalized $t$-weight space of a module of dimension $d$, and choose a lattice basis ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_r$ of $L$. On that weight space the commuting operators ${\theta }_{{\mathrm{\ell }}_i}$ are simultaneously upper-triangular with sole eigenvalues $t({\mathrm{\ell }}_i)$, by the definition of generalized weight space. Hence every $f\in A$ acts there with sole eigenvalue $f(t)$; $f-f(t)$ is nilpotent on a space of dimension at most $d$, so its $d$th power kills the initial weight space. For invariant $f$, centrality then makes the same power kill all of $U$. Any weight $t^{\mathrm{\prime }}$ of $U$ has $f(t^{\mathrm{\prime }})=f(t)$ for every invariant, forcing $t^{\mathrm{\prime }}\in W_{P}t$ by separation.

If $V$ is tempered, every logarithmic weight belongs to $K_P$. For $w\in W^P$, $w{K}_P\subset K_B$ because $w(P)$ is positive. The multiset formula proves $I_P(V)$ tempered. Conversely the injected copy of $V$ has logarithmic weights in $K_B$ if $I_P(V)$ is tempered; this alone need not give $K_P$, so additionally apply the formula for the longest minimal representative $w_{\ast }=w_0{w}_{P,0}\in W^P$. Writing a logarithmic weight $\lambda$ as central part plus simple-coroot coefficients, the conditions $\lambda \in K_B$ and $w_{\ast }\lambda \in K_B$ imply $\lambda \in K_P$: pair $\lambda$ with every dominant vector $z\in E^P\cap C$, noting that $w_{\ast }z=w_{0}z$ is antidominant, to obtain both signs of the same pairing, and use also all global central vectors. More directly, $\lambda \in K_B$ gives nonpositive simple-coroot coefficients; for $z$ in the relative interior of $E^P\cap C$, $w_{\ast }^{-1}{w}_{0}z=w_{P,0}z=z$, so $(w_{\ast }\lambda ,w_{0}z)=(\lambda ,z)$. The left side is nonnegative because $w_{\ast }\lambda \in K_B$ and $w_{0}z$ is antidominant, while $(\lambda ,z)\le 0$. Equality forces every coefficient outside $P$ to vanish. Thus $\lambda \in K_P$. The existence of this $w_{\ast }$ and its positivity on $P$ follows because $w_{P,0}$ reverses subsystem positives and $w_0$ reverses all positives.

For $\nu \in E^P$ the character $e^{\nu }$ is fixed by $W_P$ and takes value one on every root $\beta \in P$. Consequently scaling ${\theta }_{\mathrm{\ell }}$ by $e^{\nu (\mathrm{\ell })}$ preserves the lattice, finite Hecke, and both ordinary and doubled-root cross-relations: every coefficient root monomial in those relations is fixed, and the two numerator terms acquire the same scalar. Replacing $\nu$ by $-\nu$ is the inverse. Twisting a module by ${\psi }_{\nu }$ adds $\nu$ to all its logarithmic weights, so the last assertion follows from the definitions.