Top-restriction lemma and proof

This lemma proves H5 at one orbit, for a dominant lift of its label, from four inputs S1–S4. Proposition III supplies them for every symmetric space with connected \(Z_{G^\theta}(S)\).

Setting

Work in the setting of Proposition I. Fix an orbit label and a dominant lift of it; \(j\) includes the orbit in its closure, and \(i\) includes the boundary:

$$\mu\in\Lambda_S^+,\qquad\lambda\in\varphi^{-1}(\mu),\qquad j:\mathcal O_\mu\hookrightarrow\overline{\mathcal O_\mu},\qquad i:\overline{\mathcal O_\mu}\setminus\mathcal O_\mu\hookrightarrow\overline{\mathcal O_\mu}.$$

The integer \(d_\nu\) is the normalization shift of IC complexes on \(\mathcal O_\nu\), so that \(\mathrm{IC}(\overline{\mathcal O_\nu},\mathcal L)\) restricts to \(\mathcal L[d_\nu]\) on \(\mathcal O_\nu\).

Inputs

  • S1 · Finite IC decomposition. The convolution is a finite sum of shifted IC complexes on orbit closures below \(\mu\), with nonzero equivariant coefficients: $$V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{b\in A}\mathrm{IC}(\overline{\mathcal O_{\nu_b}},\mathcal L_b)[s_b]\quad\text{in }\mathcal C,\qquad\#A\lt\infty,\qquad\nu_b\le\mu.$$
  • S2 · Restriction to the orbit. On \(\overline{\mathcal O_\mu}\), restriction \(a_\mu^*=j^*\) is exact, vanishes on objects supported on the boundary, and restricts each IC complex of the top closure to its coefficient: $$j^*i_*=0,\qquad j^*\mathrm{IC}(\overline{\mathcal O_\mu},\mathcal L)\simeq\mathcal L[d_\mu].$$
  • S3 · Constant top coefficients. Each coefficient on the top orbit is a finite nonzero sum of constant sheaves: $$\mathcal L_b\simeq\underline k_{\mathcal O_\mu}^{\oplus m_b}\quad(\nu_b=\mu),\qquad m_b\in\mathbb Z_{\gt0}.$$
  • S4 · Top occurrence. Some summand lives on the top orbit closure: $$\exists\,b\in A:\quad\nu_b=\mu.$$

Lemma · Top restriction

Claim. Under S1–S4, \(\lambda\) satisfies H5 at \(\mu\):

$$\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_\mu},$$ $$a_\mu^*(V_\lambda\ast\mathrm{IC}_0)\simeq\bigoplus_{b:\,\nu_b=\mu}\underline k_{\mathcal O_\mu}[d_\mu+s_b]^{\oplus m_b}\quad\text{in }\mathcal C_\mu,\qquad0\lt N_\mu=\sum_{b:\,\nu_b=\mu}m_b\lt\infty.$$

The deduction S1–S4 ⇒ H5 uses no symmetry; symmetry enters only through the inputs.[T]

For a symmetric space. H2 and the definition of IC as \(j_{!*}\) give S2, and H3 gives S3 through connected stabilizers (Proposition III, step 4); H3 holds exactly when \(Z_{G^\theta}(S)\) is connected. S1 comes from the decomposition theorem and the support bound \(\nu_b\le\varphi(\lambda)\), and S4 from a nonzero stalk at \(t^\lambda x_0\in\mathcal O_{\varphi(\lambda)}\); these two inputs are where \(\varphi(\lambda)=\mu\) is used, and both hold over \(\mathbb C\) once their ℓ-adic proofs are transferred ().

Sources

References

References checked 7 October 2026. The reader in D holds located extracts.

Proof

Steps 1–4 restrict the decomposition S1 to the orbit, and steps 5–7 identify what survives.

  1. By S1 the convolution lives on \(\overline{\mathcal O_\mu}\), which is H5(a), and there \(a_\mu^*\) is \(j^*\). We restrict the decomposition S1 to the orbit:

    $$a_\mu^*(V_\lambda\ast\mathrm{IC}_0)=j^*\Big(\bigoplus_{b\in A}\mathrm{IC}(\overline{\mathcal O_{\nu_b}},\mathcal L_b)[s_b]\Big)\in\mathcal C_\mu.$$

  2. A summand supported on a smaller closure is disjoint from the orbit, so by S2 its restriction vanishes:

    $$\nu_b\lt\mu\quad\Longrightarrow\quad j^*\mathrm{IC}(\overline{\mathcal O_{\nu_b}},\mathcal L_b)=0.$$
    Proof
    • By H1, \(\overline{\mathcal O_{\nu_b}}\) lies in the boundary \(\overline{\mathcal O_\mu}\setminus\mathcal O_\mu\), so the summand is \(i_*\) of an object there, and \(j^*i_*=0\).

  3. A summand on the top closure restricts to its coefficient:

    $$\nu_b=\mu\quad\Longrightarrow\quad j^*\mathrm{IC}(\overline{\mathcal O_\mu},\mathcal L_b)[s_b]\simeq\mathcal L_b[d_\mu+s_b].$$
    Proof
    • \(\mathrm{IC}(\overline{\mathcal O_\mu},\mathcal L_b)=j_{!*}(\mathcal L_b[d_\mu])\) is the image of \({}^pj_!\to{}^pj_*\). Since \(j^*\) is t-exact and \(j^*j_!\simeq\mathrm{id}\simeq j^*j_*\), it sends this image to \(\mathcal L_b[d_\mu]\).

  4. Restriction is additive, so only the top summands survive:

    $$a_\mu^*(V_\lambda\ast\mathrm{IC}_0)\simeq\bigoplus_{b:\,\nu_b=\mu}\mathcal L_b[d_\mu+s_b].$$

  5. S3 turns each surviving coefficient into constant sheaves:

    $$a_\mu^*(V_\lambda\ast\mathrm{IC}_0)\simeq\bigoplus_{b:\,\nu_b=\mu}\underline k_{\mathcal O_\mu}[d_\mu+s_b]^{\oplus m_b}.$$
    Proof
    • Under H3 the stabilizers are connected, so every equivariant local system on \(\mathcal O_\mu\) is a sum of copies of \(\underline k_{\mathcal O_\mu}\). Proposition III shows that H3 holds exactly when \(Z_{G^\theta}(S)\) is connected.

  6. S4 holds for a symmetric space: the stalk at \(x_\mu=t^\lambda x_0\) is nonzero in its lowest degree, so some summand lives on the top closure.

    $$H^{-d}\big((V_\lambda\ast\mathrm{IC}_0)_{x_\mu}\big)\supseteq H^0(F,k)\ne0,\qquad F\ni t^\lambda.$$
    Proof
    • Proper base change computes the stalk on the projective fibre, with coefficients in \(V_\lambda\): $$F=\{g\in\operatorname{Gr}_{\le\lambda}:g^{-1}x_\mu\in L^+X\},\qquad d=\langle2\rho,\lambda\rangle.$$
    • \(V_\lambda\) lives in degrees \(\ge-d\), and \(\mathcal H^{-d}V_\lambda=j_*\underline k_{\operatorname{Gr}_\lambda}\supseteq\underline k\) by Deligne's formula, as \(\operatorname{Gr}_\lambda\) is dense. So \(H^{-d}(F,V_\lambda)=H^0(F,\mathcal H^{-d}V_\lambda)\supseteq H^0(F,k)\), which is nonzero since \(t^\lambda\in F\).[T]
    • Remark (weights). With \(U=F\cap\operatorname{Gr}_\lambda\) and \(e=\dim U\), \(H^{2e}_c(U)\) has one line per \(e\)-dimensional component, so it is nonzero; it is pure of weight \(2e\), and \(H^{2e-d-1}(F\setminus U,V_\lambda)\) has weights \(\le2e-1\), so \(0\ne H^{2e}_c(U)\hookrightarrow H^{2e-d}\) of the stalk. This gives the same conclusion with more input.
    • Only summands with \(\nu_b=\mu\) contribute to the stalk at \(x_\mu\in\mathcal O_\mu\), so one of them exists.

  7. S4 makes the index set nonempty and S1 makes it finite, which is H5(b) at \(\mu\) for the lift \(\lambda\); with H5(a) from step 1, this is H5:

    $$0\lt N_\mu=\sum_{b:\,\nu_b=\mu}m_b\lt\infty.$$