1982 · Perverse sheaves. BBD 4.2.5–4.3.1 supplies the smooth-pullback, intermediate-extension and finite-length results used in IIa.

1987 · Trace functions. Laumon §1.1, (1.1.1.1)–(1.1.1.3), packages additivity and the trace formula used in IIb.

2012 · Equivariant decomposition. Sun's Theorem 1.2 supplies a quotient-stack version of geometric decomposition, with the properness and purity conditions stated below; this keeps the summands equivariant.

• Groups, the Iwahori subgroup and the affine Weyl group

  \(p\) is an odd prime, \(q\) a power of \(p\), \(k=\overline{\mathbb F}_q\), \(\ell\ne p\). \(G\supseteq B\supseteq T\) is a split connected reductive group over \(\mathbb F_q\) and \(X\) a smooth affine \(G\)-variety over \(\mathbb F_q\), a row of Table 1 of the verdict.

  $$LG\supseteq K=L^+G\supseteq I,\qquad\widetilde W=X_*(T)\rtimes W=W_{\mathrm{aff}}\rtimes\Omega,\qquad\mathrm{Fl}=LG/I=\bigsqcup_{w\in\widetilde W}IwI/I .$$

  \(I\) is the preimage of \(B\) under \(L^+G\to G\), \(\ell\) the length and \(\le\) the Bruhat order on \(\widetilde W\); \(\mathrm{Fl}_{\le w}=\overline{IwI/I}\) is projective.

• Levels, pieces and the sheaf categories

  Write \(i_N:Y_N\hookrightarrow LX\) for the closed inclusion. The full normalized level functor is \(\Phi_{N,M}=i_{N,!}\pi_{N,M}^*[e_M]\); the extension by zero is part of the functor.

  A level presentation of the loop space, with \(I\) acting on \(Y_{N,M}\) through \(I/I_{N,M}\):

  $$LX=\varinjlim_NY_N,\qquad Y_N=\varprojlim_MY_{N,M},\qquad\pi_{N,M}:Y_N\to Y_{N,M}.$$

  Pieces. A point of \(Y_N\) is an infinite expansion, and \(I\) has infinitely many orbits on \(Y_N\) in general (for \(\mathbb A^1\): the \(t^k\mathcal O^\times\), \(k\ge-N\), and \(\{0\}\)). The level \((N,M)\) keeps only the image under \(\pi_{N,M}\), on which the finite-type group \(H_{N,M}=I/I_{N,M}\) has finitely many orbits (I1). A piece of level \((N,M)\) is the set of points of \(Y_N\) whose image lies in one such orbit:

  $$P=\pi_{N,M}^{-1}(o)=\{y\in Y_N:\ \pi_{N,M}(y)\in o\},\qquad o\subseteq Y_{N,M}\ \text{an }H_{N,M}\text{-orbit},$$ $$\mathcal P_{N,M}=\big\{\pi_{N,M}^{-1}(o):\ o\in H_{N,M}\backslash Y_{N,M}\big\},\qquad P'\le P\iff P'\subseteq\overline P=\pi_{N,M}^{-1}(\bar o).$$

  So a piece is an \(I\)-stable subset of \(Y_N\) cut out by conditions on finitely many coordinates. It is a union of \(I\)-orbits, not always one: for \(\mathbb A^1\) at level \((0,M)\) the pieces are the orbits \(t^j\mathcal O^\times\), \(0\le j\lt M\), and \(t^M\mathcal O\), which contains the infinitely many orbits \(t^k\mathcal O^\times\), \(k\ge M\), and \(\{0\}\) (example).

  Lemma · Pieces partition \(Y_N\). At every level \((N,M)\) the pieces are nonempty, \(I\)-stable and pairwise disjoint, \(Y_N=\bigsqcup_{P\in\mathcal P_{N,M}}P\), and \(o\mapsto\pi_{N,M}^{-1}(o)\) is a bijection \(H_{N,M}\backslash Y_{N,M}\to\mathcal P_{N,M}\); in particular \(|\mathcal P_{N,M}|=\#(H_{N,M}\backslash Y_{N,M})\).

  Proof

  • Nonempty. \(\pi_{N,M}\) is surjective, a limit of the surjective affine-space fibrations \(Y_{N,M'}\to Y_{N,M}\) of (Lev); so \(\pi_{N,M}^{-1}(o)\ne\varnothing\).
  • \(I\)-stable. \(\pi_{N,M}\) is \(I\)-equivariant and \(I\) acts on \(Y_{N,M}\) through \(H_{N,M}\): if \(\pi_{N,M}(y)\in o\) and \(g\in I\), then \(\pi_{N,M}(gy)=\bar g\,\pi_{N,M}(y)\in o\).
  • Partition. The orbits partition \(Y_{N,M}\), and taking preimages preserves disjoint unions: \(Y_N=\pi_{N,M}^{-1}(Y_{N,M})=\bigsqcup_o\pi_{N,M}^{-1}(o)\).
  • Bijection. Onto by definition of \(\mathcal P_{N,M}\); injective because \(\pi_{N,M}(\pi_{N,M}^{-1}(o))=o\) by surjectivity.

  Refinement. For \(M'\ge M\), each piece of level \((N,M')\) lies in exactly one piece of level \((N,M)\), and \(P\) is the union of those it contains. For \(N'\ge N\) and \(M\gg0\), the pieces of level \((N,M)\) are the pieces of level \((N',M)\) contained in \(Y_N=\pi_{N',M}^{-1}(Y_{N,M})\).

  $$\mathcal C_I^{(N,M)}=\pi_{N,M}^*D^b_c\big(I/I_{N,M}\backslash Y_{N,M}\big),\qquad\mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)},\qquad G_w=\mathrm{IC}_w\ast_I\mathrm{IC}_0,\qquad\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{G_w:w\in\widetilde W\}.$$

  Here \(\mathcal H_{c,I}\) is the Iwahori–Hecke category and \(\mathcal G\ast_IM=m_!(\mathcal G\mathbin{\widetilde\boxtimes}M)\).

  The shift \(e_M\). \(\pi_{N,M}\) has infinite-dimensional fibres, so \(Y_N\) has no relative dimension over \(Y_{N,M}\); the perverse normalization is fixed level by level by an integer \(e_M\). With \(f:Y_{N,M'}\to Y_{N,M}\) the transition (\(M\le M'\)), of fibre rank \(d(M,M')\), \(f^*[d(M,M')]\) is t-exact, and the two embeddings of a perverse \(K\) agree exactly when

  $$\Phi_{N,M}=i_{N,!}\pi_{N,M}^*[e_M],\qquad\Phi_{N,M'}\big(f^*K[d(M,M')]\big)=\Phi_{N,M}(K)\iff e_M=e_{M'}+d(M,M').$$

  Fix \(e_{M_0}=0\) at one level; the relation determines every \(e_M\), and by (Dict)(a) it does not depend on \(N\). For \(\mathbb A^n\): \(d(M,M')=n(M'-M)\), so \(e_M=-nM\) with \(e_0=0\).

  Simple objects. Let \(\mathcal P_I\) be the glued perverse heart and \(\mathcal S_I\) the set of isomorphism classes of its simple objects. For \(P=\pi_{N,M}^{-1}(o)\in\mathcal P_{N,M}\), with \(\Phi_{N,M}=i_{N,!}\pi_{N,M}^*[e_M]:D^b_c([Y_{N,M}/H_{N,M}])\to\mathcal C_I\) the level functor,

  $$\mathrm{IC}(P)=\Phi_{N,M}\big(\mathrm{IC}(\bar o)\big),\qquad\mathrm{IC}(P)|_P=\overline{\mathbb Q}_{\ell,P}[\delta(P)],\qquad\delta(P)=\dim o+e_M .$$

  A piece and its open dense piece \(P^\circ\) at a finer level give the same object. By I2, with \(\sim\) generated by \(P\sim P^\circ\),

  $$\mathcal P_\infty=\Big(\bigsqcup_{N,M}\mathcal P_{N,M}\Big)\Big/\!\sim\ \xrightarrow{\ \sim\ }\ \mathcal S_I,\qquad[P]\longmapsto\mathrm{IC}(P).$$

  The bijection is proved in Lemma IIa.

  For \(\mathbb A^1\), \(e_0=0\): \(\delta(t^j\mathcal O^\times)=(M-j)-M=-j\), \(\delta(t^M\mathcal O)=-M\), \(\mathcal P_\infty\simeq\mathbb Z\) and \(\mathcal S_I=\{\underline{\overline{\mathbb Q}}_{\ell,t^j\mathcal O}[-j]:j\in\mathbb Z\}\); in particular \(\mathrm{IC}_0=\mathrm{IC}(\mathcal O^\times)\) is perverse. For \(\mathbb A^n\): \(\delta(O_{m,j})=-(nm+n-j)\) (Proposition VI).

• Functions over \(F=\mathbb F_q(\!(t)\!)\)

  \(F=\mathbb F_q(\!(t)\!)\), \(\mathcal O=\mathbb F_q[\![t]\!]\), \(I(\mathbb F_q)\subseteq G(F)\) the Iwahori subgroup, Haar measure with \(\operatorname{vol}I=1\):

  $$\mathcal H_I=C_c(I\backslash G(F)/I),\quad T_w=\mathbf 1_{IwI},\qquad C_I=C_c^\infty(X(F))^I,\quad\varphi_0=\mathbf 1_{X(\mathcal O)},$$ $$(f\ast\varphi)(x)=\sum_{gI\in G(F)/I}f(g)\,\varphi(g^{-1}x).$$

  Iwahori generation of functions.

  $$\text{(IG)}\qquad\mathcal H_I\ast\varphi_0=C_I .$$

  The proposition uses \(\overline{\mathbb Q}_\ell\)-valued functions. Generation with \(\mathbb C\)-valued functions is equivalent: the piece indicators and \(T_w\ast\varphi_0\) are integer-valued, so every finite spanning problem has the same rank over any characteristic-zero coefficient field.

• Weil structures and traces

  A Weil object is \((M,\theta)\), \(M\in\mathcal C_I\), \(\theta:\mathrm{Fr}^*M\xrightarrow{\sim}M\), with \(\mathrm{Fr}\) the \(q\)-Frobenius. For \(x\in LX(\mathbb F_q)=X(F)\) and a geometric point \(\bar x\) over it:

  $$\operatorname{tr}(M)(x)=\sum_i(-1)^i\operatorname{tr}\big(\theta_{\bar x},\ \mathcal H^i(M)_{\bar x}\big)\in\overline{\mathbb Q}_\ell .$$

  The trace is additive on Weil-compatible triangles and satisfies \(\operatorname{tr}(M[1])=-\operatorname{tr}M\). Each simple \(S=\mathrm{IC}(P)\) has the canonical Weil structure restricting to the identity on \(\overline{\mathbb Q}_\ell[\delta(P)]\); the generators carry the induced Weil structures. Only the first Frobenius is used.

• Functions at one level and the piece indicators

  For a level \((N,M)\), \(C_I^{(N,M)}\) is the space of \(I\)-invariant functions supported in \(Y_N(\mathbb F_q)\) and constant on the fibres of \(\pi_{N,M}\). For a piece \(P=\pi_{N,M}^{-1}(o)\):

  $$P(\mathbb F_q)=Y_N(\mathbb F_q)\cap\pi_{N,M}^{-1}\big(o(\mathbb F_q)\big),\qquad\mathbf 1_{P(\mathbb F_q)}\in C_I^{(N,M)} .$$

  Under I2 and (Dict)(c) the \(\mathbf 1_{P(\mathbb F_q)}\), \(P\in\mathcal P_{N,M}\), are a \(\mathbb Z\)-basis of the integer-valued functions in \(C_I^{(N,M)}\), and \(C_I=\bigcup_{N,M}C_I^{(N,M)}\) (step 3).

• The IC-trace basis of \(C_I\)

  The traces of the simple objects, with canonical Weil structures:

  $$\{\operatorname{tr}S\}_{S\in\mathcal S_I},\qquad\operatorname{tr}\mathrm{IC}(P)=(-1)^{\delta(P)}\mathbf 1_{P(\mathbb F_q)}+\sum_{P'\lt P}c_{P'P}\mathbf 1_{P'(\mathbb F_q)} .$$

  They form a basis of \(C_I\), triangular against the piece indicators at a common level (step 5). \(\operatorname{tr}S\) is not an indicator: it is generally nonzero on lower pieces.

• Multiplicity spaces and their trace coefficients

  With \(P_i={}^p\mathcal H^i(G_w)\) and its induced Weil structure \(\theta_i\), the multiplicity space of \(S\) in degree \(i\) and its Frobenius are

  $$E_{S,i}(w)=\operatorname{Hom}(S,P_i),\qquad A_{S,i}(w)f=\theta_i\circ\mathrm{Fr}^*(f)\circ\theta_S^{-1}.$$

  The matrix has rows indexed by simple objects and columns by \(w\in\widetilde W\):

  $$m_{S,w}=\sum_i(-1)^i\operatorname{tr}A_{S,i}(w).$$

  Only finitely many pairs \((S,i)\) occur for each \(w\). A nonzero coefficient forces a nonzero multiplicity space.

These are the assumptions used below; (DT) is an equivariant geometric splitting, with a sufficient finite-type model supplied by Sun’s theorem.

• (Lev) · Level axioms.

  In the level presentation: each \(Y_{N,M'}\to Y_{N,M}\) (\(M'\ge M\)) is a Zariski-locally trivial affine-space fibration, equivariant for \(I/I_{N,M'}\to I/I_{N,M}\); each \(Y_N\subseteq Y_{N+1}\) is a closed embedding with \(Y_N=\pi_{N+1,M}^{-1}(Y_{N,M})\) for \(M\gg0\); each \(I/I_{N,M}\) is a connected linear algebraic group and \(I_{N,M}\) is pro-unipotent; and $$\pi_{N,M}^*:\ D^b_c\big(I/I_{N,M}\backslash Y_{N,M}\big)\longrightarrow\mathcal C_I\quad\text{is fully faithful with image }\mathcal C_I^{(N,M)} .$$

  Example · \(X=\mathbb A^1\), \(G=\mathrm{GL}_1\)

  Here \(B=T=G\), so \(I=L^+\mathrm{GL}_1=\mathcal O^\times\), acting on \(LX=F\) by multiplication. For \(N,M\ge0\) with \(N+M\ge1\):

  $$Y_N=t^{-N}\mathcal O,\qquad Y_{N,M}=t^{-N}\mathcal O/t^M\mathcal O\cong\mathbb A^{N+M},\quad\textstyle\sum_{i\ge-N}a_it^i\longmapsto(a_{-N},\dots,a_{M-1}),$$ $$I_{N,M}=1+t^{N+M}\mathcal O,\qquad I/I_{N,M}=(\mathcal O/t^{N+M})^\times\cong\mathbb G_m\times\mathbb A^{N+M-1}.$$

  • Kernel. \((1+t^{N+M}c)v-v\in t^{N+M}\cdot t^{-N}\mathcal O=t^M\mathcal O\), so \(I_{N,M}\) acts trivially; it is pro-unipotent, and \(I/I_{N,M}\) is connected (\(u_0\ne0\), \(u_1,\dots,u_{N+M-1}\) free).
  • Transitions. \(Y_{N,M'}\to Y_{N,M}\) forgets \(a_M,\dots,a_{M'-1}\): a trivial, equivariant affine-space bundle of rank \(d(M,M')=M'-M\).
  • Embeddings. \(Y_N\subseteq Y_{N+1}\) is the closed subset \(\{a_{-N-1}=0\}\), and \(Y_N=\pi_{N+1,M}^{-1}(Y_{N,M})\) for every \(M\).
  • Full faithfulness. Pullback along an affine-space bundle \(f\) is fully faithful (\(f_*f^*\simeq\mathrm{id}\)), and \(\pi_{N,M}\) is a limit of such.

  \(X=\mathbb A^n\), \(G=\mathrm{GL}_n\): \(Y_{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n\cong\mathbb A^{n(N+M)}\), \(I_{N,M}=1+t^{N+M}M_n(\mathcal O)\), \(d(M,M')=n(M'-M)\). The level \(N=M=0\) is excluded: its kernel would be all of \(I\), which is not pro-unipotent.

  Where it holds

  For \(\mathbb A^n\): \(Y_{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n\), \(I_{N,M}=1+t^{N+M}M_n(\mathcal O)\). For the group: Bruhat closures and congruence subgroups. The congruence choice is needed: for \(\mathrm{SL}_2\) the full kernel contains \(\Delta\mu_2\), which is disconnected (A7-5).

• I1 · Finite levels.

  \(LX\) has a level presentation \(LX=\varinjlim_NY_N\), \(Y_N=\varprojlim_MY_{N,M}\), such that every object comes from one level and every level has finitely many pieces: $$\text{(i)}\quad\forall\,\mathcal F\in\mathcal C_I\ \ \exists\,(N,M),\ \mathcal F_0\in D^b_c\big(I/I_{N,M}\backslash Y_{N,M}\big):\quad\mathcal F\simeq\pi_{N,M}^*\mathcal F_0\ \ \text{(extended by zero from }Y_N),$$ $$\text{(ii)}\quad\forall\,(N,M):\quad|\mathcal P_{N,M}|=\#\big((I/I_{N,M})\backslash Y_{N,M}\big)<\infty .$$

  Equivalently \(\mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)}\) and \(|\mathcal P_{N,M}|<\infty\); \(\mathcal P_{N,M}\) is defined in Levels and pieces. These two conditions are exactly what step 7 uses: (i) puts an object at one level, (ii) gives that level's perverse heart finite length.

  Example · \(X=\mathbb A^1\), \(G=\mathrm{GL}_1\)

  With the levels of the (Lev) example, the orbits of \((\mathcal O/t^{N+M})^\times\) on \(Y_{N,M}\) and the pieces are

  $$o_j=\{a_{-N}=\dots=a_{j-1}=0\ne a_j\}\cong\mathbb G_m\times\mathbb A^{M-j-1}\ \ (-N\le j<M),\qquad o_M=\{0\},$$ $$P_j=\pi_{N,M}^{-1}(o_j)=t^j\mathcal O^\times,\qquad P_M=\pi_{N,M}^{-1}(0)=t^M\mathcal O,\qquad P_M<P_{M-1}<\dots<P_{-N},\qquad|\mathcal P_{N,M}|=N+M+1.$$

  • One orbit. \(v=t^ju\) with \(u\) a unit, determined mod \(t^{M-j}\), and \((\mathcal O/t^{N+M})^\times\to(\mathcal O/t^{M-j})^\times\) is onto because \(M-j\le N+M\).
  • Closures. \(\overline{o_j}=t^j\mathcal O/t^M\mathcal O\cong\mathbb A^{M-j}\), so \(\overline{P_j}=t^j\mathcal O\): the pieces form a chain.
  • Refinement. \(M\to M+1\) splits \(P_M=t^M\mathcal O\) into \(t^M\mathcal O^\times\) and \(t^{M+1}\mathcal O\); \(N\to N+1\) adds \(t^{-N-1}\mathcal O^\times\).

  \(\mathbb A^n\) under \(\mathrm{GL}_n\): \(|\mathcal P_{N,M}|=(N+M)n+1\), a chain (III).

• I2 · Connected stabilizers.

  Every stabilizer of \(I/I_{N,M}\) on \(Y_{N,M}\) is connected; so every \(I\)-equivariant local system on a piece is constant.

  Where it is used

  Used to identify simple equivariant perverse sheaves with constant-local-system IC sheaves and to obtain one rational orbit on each finite-level orbit. These inputs give the IC-trace basis at the first Frobenius.

• (Dict) · Trace dictionary over \(\mathbb F_q\).

  1. Rationality. \(G\supseteq B\supseteq T\) are split over \(\mathbb F_q\), \(q\) odd; \(X\), the action, the level presentation and its transition maps are defined over \(\mathbb F_q\), so that \(LX(\mathbb F_q)=X(F)\); the ranks \(d(M,M')\) of \(Y_{N,M'}\to Y_{N,M}\) do not depend on \(N\).
  2. Topology. \(Y_N(\mathbb F_q)\) is compact open in \(X(F)\), \(\bigcup_NY_N(\mathbb F_q)=X(F)\), and the fibres of \(\pi_{N,M}\) on \(Y_N(\mathbb F_q)\), \(M\ge0\), form a basis of the topology of \(Y_N(\mathbb F_q)\).
  3. Rational orbits. Every \(I/I_{N,M}\)-orbit on \(Y_{N,M}\) is Frobenius-stable.
  4. Bounded convolution. For \(w\in\widetilde W\) and \(N\) there is \(N'\) with \(\overline{IwI}\cdot Y_N\subseteq Y_{N'}\), and for each \(M\) an \(M''\) such that \((g,x)\mapsto\pi_{N,M}(g^{-1}x)\), on \(\{(g,x)\in\overline{IwI}\times Y_{N'}:g^{-1}x\in Y_N\}\), factors through \(\overline{IwI}\times Y_{N',M''}\).

  Sources

  Cited theorems and where it holds

  Lang’s theorem, the trace formula and the finite-level perverse structures are used in steps 1–5 below and in the trace-of-convolution lemma. For (DT), the proper quotient-stack model must also carry a pure convolution kernel, as specified after Sun’s theorem. Boundedness alone is not a substitute for these hypotheses.

• (DT) · Equivariant geometric decomposition.

  For every \(w\), the generator is a finite sum of shifted simple perverse objects in \(\mathcal C_I\), over \(k=\overline{\mathbb F}_q\):

  $$G_w\simeq\bigoplus_{S\in\mathcal S_I,\ i\in\mathbb Z}S[-i]\otimes E_{S,i}(w).$$

  A proper, pure finite-type quotient-stack model gives this by Sun’s Theorem 1.2 below. The splitting retains \(I\)-equivariance; Frobenius acts on the multiplicity spaces and need not preserve a chosen derived splitting.

For each \(w\), a sufficient model for (DT) is a proper finite-diagonal map \(f_{w,0}:\mathscr Z_{w,0}\to\mathscr Y_{w,0}\) between finite-type quotient stacks with affine stabilizers, together with a pure kernel \(K_{w,0}\), whose push-forward realizes \(G_w\) at one level:

Here \(\Phi_w\) is the fully faithful level pullback followed by extension by zero, with the fixed perverse normalization. Sun’s theorem gives (DT) on the quotient stack, so the summands retain equivariance. See proper-model criterion for the model criterion.

The splitting is geometric and need not be Frobenius-equivariant. Perverse cohomology still carries Frobenius; its multiplicity spaces supply the trace coefficients in step 6.