§1 · Agent 1 · Literature and Table 1 audit · 2026-10-18T18:42:30Z

Lemma 1.2 · The group \(H\) under \(H\times H\): every hypothesis checked, \(M_I\) is the lowest cell

proved ★ key step

For the group all inputs of the Iwahori summand criterion and of the trace bridge hold (finite Bruhat levels, connected stabilizers, rational orbits, parity decomposition), and the set of summands of the generators is the lowest two-sided cell, which omits the unit when \(W_H\neq1\).

Lemma 1.2.

Let \(X=H\) under \(G=H\times H\), \((h_1,h_2)\cdot x=h_1xh_2^{-1}\), \(W_H\ne1\); \(K=K_H^2\), \(I=I_H^2\), \(\widetilde W\) the extended affine Weyl group of \(H\), \(w_0\) the longest element of \(W_H\), \(c=\underline k_{K_H/I_H}\). Over \(\overline{\mathbb F}_q\) (\(\ell\)-adic) and over \(\mathbb C\) (Betti):

  1. (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).) \(Y_N=\bigcup_{\ell(w)\le N}\overline{I_HwI_H}\), \(Y_{N,M}=Y_N/I_M\); the pieces of every level are the cells \(I_HvI_H\), finitely many.
  2. (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) Every stabilizer of \(I/I_{N,M}\) on \(Y_{N,M}\) is connected.
  3. (Frob) Every cell is defined over \(\mathbb F_q\) and \(I_H(\mathcal O)\backslash H(F)/I_H(\mathcal O)=\widetilde W\) for every non-archimedean \(F\).
  4. (\(\mathrm{IC}_0\)) \(L^+X=K_H\) is one \(K\)-orbit, closed and pro-smooth; \(\mathrm{IC}_0=c\simeq\mathrm{IC}_{w_0}[-\ell(w_0)]\).
  5. The simple objects of \(\mathcal C_I=\mathcal H_{c,I_H}\) are the \(\mathrm{IC}_z\), \(z\in\widetilde W\); \(\mathcal H_{c,I}=\operatorname{Thick}\{\mathrm{IC}_a\boxtimes\mathrm{IC}_b\}\) and \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_a\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_b\}\).
  6. (DT with parity) \(\mathrm{IC}_a\ast\mathrm{IC}_b\simeq\bigoplus_z\bigoplus_n\mathrm{IC}_z[n]^{\oplus m_{z,n}}\) with \(m_{z,n}=0\) unless \(n\equiv\ell(a)+\ell(b)-\ell(z)\pmod 2\).
  7. \(M_I:=\{z:\mathrm{IC}_z[s]\text{ is a summand of some }\mathrm{IC}_a\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_b\}=\{z:z\le_{LR}w_0\}\), and \(e\notin M_I\).

Proof.

  1. (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).). Bruhat decomposition \(LH=\bigsqcup_{\widetilde W}I_HwI_H\) (Iwahori–Matsumoto, Prop. 2.4 and Thm 2.5, iwahori-matsumoto-1965.pdf; for the loop group over \(k\), same proof) and finiteness of Bruhat lower sets. \(I_M=\ker(I_H\to H(\mathcal O/t^M))\) is normal in \(K_H\), every cell is right \(I_M\)-stable, so cells are preimages of their images: hypothesis.html, Example B.
  2. (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.). The stabilizer of \(wI_M\in Y_N/I_M\) in \(I_H\times I_H\) is \(\{(a,b):awb^{-1}\in wI_M\}=\{(a,\,w^{-1}aw\,u):a\in I_H\cap wI_Hw^{-1},\ u\in I_M\}\), an extension of \(I_M\) (pro-unipotent) by \(I_H\cap wI_Hw^{-1}=T(\mathcal O)\cdot(\text{pro-unipotent})\); both connected. The stabilizer in the finite quotient is its image, hence connected.
  3. (Frob). \(T\) split, so \(t^\lambda\) and Tits representatives of \(W_H\) lie in \(H(\mathbb F_q((t)))\); Iwahori–Matsumoto loc. cit. over any non-archimedean \(F\).
  4. \(K_H=K\cdot e\); closed since \(H\) affine; \(K_H=\varprojlim H(\mathcal O/t^m)\) with smooth transition maps. \(K_H/I_H=H/B_H\) is smooth, so \(c\) is the shifted IC sheaf; BY Lemma 3.2.7 records \(C_\Theta\simeq\mathrm{IC}_{w_\Theta}[-\ell_\Theta](-\ell_\Theta/2)\) (bezrukavnikov-yun-v2.pdf).
  5. \(I\times I\)-orbits on \(LH\) are the cells, with connected stabilizers (step 2), so the dévissage of hypothesis.html (Proposition III, step 1) gives \(\mathcal C_I=\operatorname{Thick}\{\Delta_z\}=\operatorname{Thick}\{\mathrm{IC}_z\}\) (induction on the Bruhat order, \(\Delta_z\to\mathrm{IC}_z[-\ell(z)]\to\) boundary). The same on \(LG=LH^2\), whose cells are products. An object \(\mathrm{IC}_a\boxtimes\mathrm{IC}_b\) acts by \(M\mapsto\mathrm{IC}_a\ast M\ast\mathrm{IC}_{b^{-1}}\) (inversion on the right factor; \(\mathrm{inv}^*\mathrm{IC}_b=\mathrm{IC}_{b^{-1}}\)); exactness of the action and (T2) give the generators.
  6. bezrukavnikov-yun-v2.pdf, Prop. 3.2.5 (Kac–Moody flag varieties, mixed \(\ell\)-adic, with the parity condition), extended to loop groups of split simply connected groups in BY §0.5; for reductive \(H\) the components of \(\mathrm{Fl}\) are indexed by \(\Omega\) and each is the affine flag variety of \(H_{\mathrm{sc}}^{\mathrm{der}}\), convolution by \(\Delta_\omega\) permuting them. Over \(\mathbb C\): BBD 6.2.5 gives semisimplicity; the parity is the same (it is read off from Kazhdan–Lusztig polynomials).
  7. Summands are \(\le_{LR}w_0\). \(\mathrm{IC}_a\) is a summand of \(\Delta_\omega\ast\mathrm{IC}_{s_1}\ast\cdots\ast\mathrm{IC}_{s_k}\) for a reduced expression \(a=\omega s_1\cdots s_k\) (6, top term), likewise \(\mathrm{IC}_b\). By Krull–Schmidt (\(\operatorname{End}\mathrm{IC}_z=k\)) each summand of \(\mathrm{IC}_a\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_b\) is reached from \(\mathrm{IC}_{w_0}\) by successive convolutions with one \(\mathrm{IC}_s\) or \(\Delta_\omega\) on the left or the right, keeping a summand each time. By 6 (parity, so no cancellation), \(\mathrm{IC}_y\) is a summand of \(\mathrm{IC}_s\ast\mathrm{IC}_x\) iff \(C'_y\) has nonzero coefficient in \(C'_sC'_x\), which is the elementary step of \(y\le_L x\) (Kazhdan–Lusztig 1979, §1); right steps give \(\le_R\). Hence \(z\le_{LR}w_0\). Conversely a chain \(z\le_{LR}w_0\) is a sequence of such steps, and the same Krull–Schmidt argument puts \(\mathrm{IC}_z\) as a summand of some \(\mathrm{IC}_a\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_b\).
  8. \(e\notin M_I\). Lusztig's \(a\)-function on \(\widetilde W\) satisfies \(a(y)\ge a(w)\) whenever \(y\le_{LR}w\) (Lusztig, Cells in affine Weyl groups I, 1985, Thm 5.4), \(a(w_J)=\ell(w_J)\) for the longest element of a finite parabolic subgroup, and \(a(e)=0\). So \(e\le_{LR}w_0\) would give \(0\ge\ell(w_0)\ge1\). Moreover \(a\le\nu=\ell(w_0)\) (loc. cit., Thm 7.2), so \(M_I\subseteq\{a=\nu\}=c_0\), the lowest two-sided cell (Shi 1987; Lusztig loc. cit. §8), and \(M_I=c_0\) since \(c_0\) is one two-sided cell containing \(w_0\). (Numbers of Lusztig's and Shi's results are recalled, not checked in a local PDF; only \(a(y)\ge a(w)\) for \(y\le_{LR}w\), \(a(w_0)=\ell(w_0)\) and \(a(e)=0\) are used for \(e\notin M_I\).)

Why a linear functional does not suffice (checklist 6). The sign character \(\varepsilon(T_w)=(-1)^{\ell(w)}\) kills the two-sided ideal generated by \(C'_{w_0}\), but \(M_I\) consists of retracts: \(\mathrm{IC}_e\) could a priori be a summand of a generator whose other summands have \(\varepsilon\)-values cancelling its own. Step 8 (positivity, via the \(a\)-function) or the pages' Whittaker functor \(\Phi\) (group.html) is needed; Proposition II is cited here for comparison only.

Status of the row. Sheaves: \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\), proved on group.html (D-modules over \(\mathbb C\), affine Whittaker functor; the \(\ell\)-adic version replaces \(\exp\) by Artin–Schreier, laumon-1987.pdf (1.1.3.4)); steps 5–8 give the second route once the summand criterion is available (\(\mathrm{IC}_e=\Delta_e\) is simple and \(e\notin M_I\)). Functions: (IG) fails for every non-archimedean \(F\) (classical.html, Proposition E, sign character; Iwahori–Matsumoto Thm 3.5). The two levels agree.

Builds on

  • Setup 1.1 · Setting audit: D-modules, Betti and ℓ-adic give the same \(\mathcal C_I\) and \(\ast_I\)

Built on by

  • Lemma 1.6 · Dictionary inputs at Iwahori level: rational orbits, Lang, trace bases, convolution
  • Audit 1.7 · Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument
  • Summary 1.10 · Verdict of the audit: rows 1–3 agree, the rest reduce to one module computation
Original record, as the agent wrote it at 18:42 UTC

The group \(H\) under \(H\times H\): every hypothesis checked, \(M_I\) is the lowest cell

Proof.

  1. (I1). Bruhat decomposition \(LH=\bigsqcup_{\widetilde W}I_HwI_H\) (Iwahori–Matsumoto, Prop. 2.4 and Thm 2.5, iwahori-matsumoto-1965.pdf; for the loop group over \(k\), same proof) and finiteness of Bruhat lower sets. \(I_M=\ker(I_H\to H(\mathcal O/t^M))\) is normal in \(K_H\), every cell is right \(I_M\)-stable, so cells are preimages of their images: hypothesis.html, Example B.
  2. (I2). The stabilizer of \(wI_M\in Y_N/I_M\) in \(I_H\times I_H\) is \(\{(a,b):awb^{-1}\in wI_M\}=\{(a,\,w^{-1}aw\,u):a\in I_H\cap wI_Hw^{-1},\ u\in I_M\}\), an extension of \(I_M\) (pro-unipotent) by \(I_H\cap wI_Hw^{-1}=T(\mathcal O)\cdot(\text{pro-unipotent})\); both connected. The stabilizer in the finite quotient is its image, hence connected.
  3. (Frob). \(T\) split, so \(t^\lambda\) and Tits representatives of \(W_H\) lie in \(H(\mathbb F_q((t)))\); Iwahori–Matsumoto loc. cit. over any non-archimedean \(F\).
  4. \(K_H=K\cdot e\); closed since \(H\) affine; \(K_H=\varprojlim H(\mathcal O/t^m)\) with smooth transition maps. \(K_H/I_H=H/B_H\) is smooth, so \(c\) is the shifted IC sheaf; BY Lemma 3.2.7 records \(C_\Theta\simeq\mathrm{IC}_{w_\Theta}[-\ell_\Theta](-\ell_\Theta/2)\) (bezrukavnikov-yun-v2.pdf).
  5. \(I\times I\)-orbits on \(LH\) are the cells, with connected stabilizers (step 2), so the dévissage of hypothesis.html (Proposition III, step 1) gives \(\mathcal C_I=\operatorname{Thick}\{\Delta_z\}=\operatorname{Thick}\{\mathrm{IC}_z\}\) (induction on the Bruhat order, \(\Delta_z\to\mathrm{IC}_z[-\ell(z)]\to\) boundary). The same on \(LG=LH^2\), whose cells are products. An object \(\mathrm{IC}_a\boxtimes\mathrm{IC}_b\) acts by \(M\mapsto\mathrm{IC}_a\ast M\ast\mathrm{IC}_{b^{-1}}\) (inversion on the right factor; \(\mathrm{inv}^*\mathrm{IC}_b=\mathrm{IC}_{b^{-1}}\)); exactness of the action and (T2) give the generators.
  6. bezrukavnikov-yun-v2.pdf, Prop. 3.2.5 (Kac–Moody flag varieties, mixed \(\ell\)-adic, with the parity condition), extended to loop groups of split simply connected groups in BY §0.5; for reductive \(H\) the components of \(\mathrm{Fl}\) are indexed by \(\Omega\) and each is the affine flag variety of \(H_{\mathrm{sc}}^{\mathrm{der}}\), convolution by \(\Delta_\omega\) permuting them. Over \(\mathbb C\): BBD 6.2.5 gives semisimplicity; the parity is the same (it is read off from Kazhdan–Lusztig polynomials).
  7. Summands are \(\le_{LR}w_0\). \(\mathrm{IC}_a\) is a summand of \(\Delta_\omega\ast\mathrm{IC}_{s_1}\ast\cdots\ast\mathrm{IC}_{s_k}\) for a reduced expression \(a=\omega s_1\cdots s_k\) (6, top term), likewise \(\mathrm{IC}_b\). By Krull–Schmidt (\(\operatorname{End}\mathrm{IC}_z=k\)) each summand of \(\mathrm{IC}_a\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_b\) is reached from \(\mathrm{IC}_{w_0}\) by successive convolutions with one \(\mathrm{IC}_s\) or \(\Delta_\omega\) on the left or the right, keeping a summand each time. By 6 (parity, so no cancellation), \(\mathrm{IC}_y\) is a summand of \(\mathrm{IC}_s\ast\mathrm{IC}_x\) iff \(C'_y\) has nonzero coefficient in \(C'_sC'_x\), which is the elementary step of \(y\le_L x\) (Kazhdan–Lusztig 1979, §1); right steps give \(\le_R\). Hence \(z\le_{LR}w_0\). Conversely a chain \(z\le_{LR}w_0\) is a sequence of such steps, and the same Krull–Schmidt argument puts \(\mathrm{IC}_z\) as a summand of some \(\mathrm{IC}_a\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_b\).
  8. \(e\notin M_I\). Lusztig's \(a\)-function on \(\widetilde W\) satisfies \(a(y)\ge a(w)\) whenever \(y\le_{LR}w\) (Lusztig, Cells in affine Weyl groups I, 1985, Thm 5.4), \(a(w_J)=\ell(w_J)\) for the longest element of a finite parabolic subgroup, and \(a(e)=0\). So \(e\le_{LR}w_0\) would give \(0\ge\ell(w_0)\ge1\). Moreover \(a\le\nu=\ell(w_0)\) (loc. cit., Thm 7.2), so \(M_I\subseteq\{a=\nu\}=c_0\), the lowest two-sided cell (Shi 1987; Lusztig loc. cit. §8), and \(M_I=c_0\) since \(c_0\) is one two-sided cell containing \(w_0\). (Numbers of Lusztig's and Shi's results are recalled, not checked in a local PDF; only \(a(y)\ge a(w)\) for \(y\le_{LR}w\), \(a(w_0)=\ell(w_0)\) and \(a(e)=0\) are used for \(e\notin M_I\).)

Why a linear functional does not suffice (checklist 6). The sign character \(\varepsilon(T_w)=(-1)^{\ell(w)}\) kills the two-sided ideal generated by \(C'_{w_0}\), but \(M_I\) consists of retracts: \(\mathrm{IC}_e\) could a priori be a summand of a generator whose other summands have \(\varepsilon\)-values cancelling its own. Step 8 (positivity, via the \(a\)-function) or the pages' Whittaker functor \(\Phi\) (group.html) is needed; Proposition II is cited here for comparison only.

Status of the row. Sheaves: \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\), proved on group.html (D-modules over \(\mathbb C\), affine Whittaker functor; the \(\ell\)-adic version replaces \(\exp\) by Artin–Schreier, laumon-1987.pdf (1.1.3.4)); steps 5–8 give the second route once the summand criterion is available (\(\mathrm{IC}_e=\Delta_e\) is simple and \(e\notin M_I\)). Functions: (IG) fails for every non-archimedean \(F\) (classical.html, Proposition E, sign character; Iwahori–Matsumoto Thm 3.5). The two levels agree.

Sources