diff --git a/restricted_geometric_langlands_pos_v2.txt b/restricted_geometric_langlands_pos_v2.txt
index e8b6517..a0f1107 100644
--- a/restricted_geometric_langlands_pos_v2.txt
+++ b/restricted_geometric_langlands_pos_v2.txt
@@ -129,7 +129,7 @@ let ellne p be a prime, and set E=Qbar_ Let
 X/k be a smooth projective connected curve and G/k a connected
 reductive group; such a group is split over k. Write
 check G/E for the Langlands dual group and
-mathfrak g=frakg=Lie(G).
+g=g=Lie(G).
 
 We work with the E-linear ind-constructible sheaf categories of
 [cite]. In particular Shv(X) is ind-constructible, and
@@ -175,13 +175,13 @@ which positive characteristic can complicate Lie theory.
 The following hold for G/k.
 [label=(roman*)]
  There is a nondegenerate G-invariant symmetric bilinear form
-  kappa on mathfrak gfrakg whose restriction to Lie Z(M) is
+  kappa on g whose restriction to Lie Z(M) is
   nondegenerate for every Levi subgroup M.
  Chevalley restriction k[Lie M]^Mto k[Lie T_M]^{W_M} is an
   isomorphism for every Levi M and every maximal torus T_Msubset M.
  The scheme-theoretic centraliser in G of every semisimple
-  element of mathfrak gfrakg is a Levi subgroup.
- Every nilpotent element of mathfrak gfrakg lies in Lie R_u(P) for
+  element of g is a Levi subgroup.
+ Every nilpotent element of g lies in Lie R_u(P) for
   some parabolic Psubset G, after any field extension.
 
 
@@ -189,7 +189,7 @@ The following hold for G/k.
 Conditions (i)--(iii) are cite[S14.4.1]{AGKRRV}; condition (iv) is
 cite[S D.1.1]{AGKRRV}. Both hypotheses below keep the notion of
 semisimple and nilpotent as in [cite]: semisimple elements of
-mathfrak gfrakg are exactly the ones geometrically conjugate into a
+g are exactly the ones geometrically conjugate into a
 Cartan subalgebra, nilpotent ones the ones whose adjoint orbit closure
 contains 0.
 
@@ -343,7 +343,7 @@ p and residue field k. It is strictly henselian, and it is
 excellent because it is a complete Noetherian local ring
 cite[Tag~07QS]{Stacks}.
 
-The curve X/k admits a smooth projective lift X/S.scrX/S.
+The curve X/k admits a smooth projective lift X/S.X/S.
 Concretely, smooth-lifting theory gives a smooth formal lift over
 SpfR, since the obstruction to lifting a smooth curve
 lies in H^2(X,T_X)=0; the ample line bundle O_X(1) lifts for the
@@ -360,10 +360,10 @@ The root datum of the split G gives a split reductive group over
 Z, and base change to S gives a lift, which we also denote
 G cite[Thms.~6.1.16(2) and~6.1.17]{Conrad}. Write
 [
- B= X/S),scrX/S),
+ B= X/S),X/S),
  s=Spec(k), eta=Spec(K).
 ]
-Then B= X,B_SG)scrX,B_SG) is an algebraic stack locally
+Then B= X,B_SG)X,B_SG) is an algebraic stack locally
 of finite type over S with affine diagonal
 cite[Thm.~1.2(i)--(iii)]{HallRydh}: X/SscrX/S is proper, flat and
 of finite presentation, and B_SG is locally of finite presentation
@@ -436,7 +436,7 @@ The corresponding map
 identifies Y with an open-and-closed union of components of Y_K
 cite[S3.1.3]{GR}. In particular the tensor product
 
- D_Y:=calD_Y:=QCoh(Y)underset{QCoh(Y_K)}{ ⊗ }C_K
+ D_Y:=D_Y:=QCoh(Y)underset{QCoh(Y_K)}{ ⊗ }C_K
 
 is a direct summand of C_K.
 
@@ -445,7 +445,7 @@ not merely a functor obtained by transporting L_G across the
 specialisation. Corollary~4.3.6 and S4.3.9 of [cite] show that the
 chartwise geometric nearby-cycle functor we built above restricts to
 [
- Psi_{B}: D_Y→calD_Y→ C_s,
+ Psi_{B}: D_Y→D_Y→ C_s,
 ]
 and it is QCoh(Y)-linear by cite[Thm.~3.1.6(B) and S4.3.10]{GR}.
 
@@ -462,7 +462,7 @@ Y'subset Y.
 If a connected component Csubset Y is disjoint from Y', then the
 summand
 [
- D_C:=calD_C:=QCoh(C)underset{QCoh(Y)}{ ⊗ }D_Y
+ D_C:=D_C:=QCoh(C)underset{QCoh(Y)}{ ⊗ }D_Y
 ]
 of D_Y is nonzero; Psi_{B} for every
 DinD_C; and there exists a nonzero FinD_C
@@ -499,7 +499,7 @@ D_C=0. Choose such an F.
 Compactness in C_K is the same argument: F is compact in
 D_Y, which is a summand of C_K, and the inclusion
 of a summand preserves compactness. The inclusion
-C_K↪calC_K↪Shv(B_K) preserves compactness
+C_K↪C_K↪Shv(B_K) preserves compactness
 by cite[Thm.~1.1.7]{GR}. Finally, a compact sheaf on an algebraic
 stack restricts to a bounded constructible complex on every smooth
 finite-type chart cite[S F.2.2]{AGKRRV}.
@@ -521,8 +521,8 @@ Y'=Y there is nothing to prove.
 
 Let V be a finite-dimensional representation of check G and let
 [
- mathrm H_VrmH_VcolonC_K→calC_K→
-   Shv(B_K X_K)scrX_K)
+ H_VrmH_V:C_K→C_K→
+   Shv(B_K X_K)X_K)
 ]
 be the one-leg Hecke functor, normalised so that it realises the
 spectral action on the spectral side. The correspondence and kernel
@@ -537,7 +537,7 @@ The universal objects are
 On an affine test Tto Y the restriction of E_V is the
 family of local systems obtained by applying the corresponding tensor
 functor to V. Acting by its first tensor factor and tensoring with
-the exterior product gives mathrm H_VrmH_V on the spectral side
+the exterior product gives H_VrmH_V on the spectral side
 cite[S14.3.3]{AGKRRV}.
 
 By definition of i,
@@ -567,10 +567,10 @@ descent, membership in IndLisse for propagation) are separate.
 Now apply the QCoh(Y)-action in (eq:spec-IndLisse) to F. The
 result is an object of D_C ⊗ IndLisse(X). After applying
 id ⊗ rho and the exterior-product functor, it is precisely
-mathrm H_V(F)rmH_V(F) by (eq:spec-universal-family). Elementary tensors
+H_V(F)rmH_V(F) by (eq:spec-universal-family). Elementary tensors
 generate D_C ⊗ IndLisse(X) under colimits, and bounded
 truncations build every object of Lisse(X) from
-finite-rank lisse sheaves. Therefore mathrm H_V(F)rmH_V(F) lies in the stable
+finite-rank lisse sheaves. Therefore H_V(F)rmH_V(F) lies in the stable
 colimit closure of objects
 
  DboxtimesL_K, DinD_C,
@@ -586,7 +586,7 @@ every finite-dimensional V,
 [
  Psi_T(a_K^*F)=0,
  
- Psi_T((a,ell_T)_K^*mathrm H_V(F))=0.rmH_V(F))=0.
+ Psi_T((a,ell_T)_K^*H_V(F))=0.rmH_V(F))=0.
 ]
 The same statements hold on the smooth locus of a, and after any of
 the finite trait replacements described above.
@@ -609,7 +609,7 @@ Psi_{B} therefore give
 ]
 Both pullback and specialisation are continuous, so the same vanishing
 holds for the stable colimit closure of these exterior generators. That
-closure contains mathrm H_V(F).rmH_V(F). Finite trait replacement preserves
+closure contains H_V(F).rmH_V(F). Finite trait replacement preserves
 geometric specialisation, so no new hypothesis is needed there.
 
 
@@ -642,9 +642,9 @@ Levi subgroup.
 
 The Lie-theoretic input is the following. Choose a point xin X(k)
 where A is non-nilpotent. In a formal coordinate t and bundle frame
-at x, write A=a(t),dt with a(t)inmathfrak g[[t]].frakg[[t]]. Chevalley
-restriction identifies the nilpotent cone in mathfrak gfrakg with
-chi^{-1}(chi(0)), where chicolonmathfrak gfrakgtomathfrak g/frakg/!/G is
+at x, write A=a(t),dt with a(t)ing[[t]].g[[t]]. Chevalley
+restriction identifies the nilpotent cone in g with
+chi^{-1}(chi(0)), where chicolonfrak g/g/!/G is
 the adjoint quotient; so the requirement that A be non-nilpotent at x
 becomes chi(a(0))nechi(0).
 
@@ -652,23 +652,23 @@ becomes chi(a(0))nechi(0).
 Let k and G satisfy Hypothesis~hyp:common and either
 Hypothesis~hyp:A or Hypothesis~hyp:B, and let kappa be the
 invariant symmetric form of Hypothesis~hyp:common(i). Let
-a(t)inmathfrak g[[t]]frakg[[t]] satisfy chi(a(0))nechi(0). Then after a
+a(t)ing[[t]]g[[t]] satisfy chi(a(0))nechi(0). Then after a
 change of frame by an element of G(k[[t]]), there are a Levi subgroup
 Msubset G and a cocharacter lambdacolonG_m Z(M) such
 that
 
- a(t)inmathfrak m[[t]],frakm[[t]], kappa(a(0),dlambda)ne0,
- ad_{a(0)}colonmathfrak g/frakg/mathfrak mfrakm
-  xrightarrow{ sim }mathfrak g/frakg/mathfrak m,frakm,
+ a(t)inm[[t]],m[[t]], kappa(a(0),dlambda)ne0,
+ ad_{a(0)}colong/g/m
+  xrightarrow{ sim }g/g/m,m,
 
-where mathfrak m=frakm=Lie M and dlambda=dlambda(1)inmathfrak gfrakg is the
+where m=m=Lie M and dlambda=dlambda(1)infrakg is the
 image of 1inLie(G_m)=k. The subgroup M may equal G; the
 invertibility includes the zero vector space.
 
 
 The proof has three layers. First, root-pairing and Jordan facts
 (Ssec:lie-basic): kappa pairs root vectors with their opposite
-partners, mathfrak tfrakt is nondegenerate, and every element is conjugate
+partners, t is nondegenerate, and every element is conjugate
 into the Lie algebra of a Borel subgroup. Second, one of two
 regime-specific arguments produces a central cocharacter
 (Ssec:lie-A and~Ssec:lie-B). Third, a formal smoothing
@@ -679,30 +679,30 @@ argument lifts the constant-term reduction to the full formal arc
 
 
 Fix a maximal torus Tsubset G. Set
-Λ=X_*(T), Λ^vee=X^*(T), mathfrak t=frakt=Lie T=Λ ⊗  k.
-For a root alpha write dalphacolonmathfrak tfraktto k for the
-differential of the character alpha and dalpha^veeinmathfrak tfrakt
+Λ=X_*(T), Λ^vee=X^*(T), t=t=Lie T=Λ ⊗  k.
+For a root alpha write dalphacolonfraktto k for the
+differential of the character alpha and dalpha^veeinfrakt
 for the differential of the coroot.
 
 
-The form kappa is nondegenerate on mathfrak t.frakt. For each root alpha
+The form kappa is nondegenerate on t.t. For each root alpha
 there is a scalar c_alphain k^times such that
 
- kappa(dalpha^vee,h)=c_alpha,dalpha(h)(hinmathfrak t).frakt).
+ kappa(dalpha^vee,h)=c_alpha,dalpha(h)(hint).t).
 
 In particular neither dalpha nor dalpha^vee vanishes. Every element
-of mathfrak gfrakg is conjugate into the Lie algebra of a Borel subgroup
+of g is conjugate into the Lie algebra of a Borel subgroup
 containing T, and every semisimple element is conjugate into
-mathfrak t.frakt.
+t.t.
 
 
 Proof. 
 Under the T-weight decomposition
-mathfrak g=frakg=mathfrak tfrakt ⊕ ⊕_alphamathfrak g_frakg_alpha,
-invariance of kappa makes mathfrak tfrakt orthogonal to each root space
-and makes mathfrak g_frakg_alpha pair only with mathfrak g_frakg_{-alpha}.
-Nondegeneracy of kappa on mathfrak gfrakg then gives nondegeneracy on
-mathfrak tfrakt and on each root pairing. Choose compatible root vectors
+g=g=t ⊕ ⊕_alphafrakg_alpha,
+invariance of kappa makes t orthogonal to each root space
+and makes g_alpha pair only with g_{-alpha}.
+Nondegeneracy of kappa on g then gives nondegeneracy on
+t and on each root pairing. Choose compatible root vectors
 e_alpha,e_{-alpha} with [e_alpha,e_{-alpha}]=dalpha^vee; this
 follows from the SL_2-root homomorphism and holds for every
 isogeny type cite[SS8.1.1--8.1.4]{Springer}, cite[Cor.~5.1.12, Eq.~(5.1.4)]{Conrad}.
@@ -716,19 +716,19 @@ and works in characteristic two.
 
 For the Borel-conjugation statement: consider the incidence morphism
 [
- Gtimes^Bmathfrak b→frakb→mathfrak g,frakg,
+ Gtimes^Bb→b→g,g,
  [g,x]|->Ad(g)x,
 ]
-where Bsupset T is a Borel subgroup and mathfrak b=frakb=Lie B. Its
-source is closed in (G/B)timesmathfrak g,frakg, so the morphism is proper.
-Choose cinmathfrak tfrakt outside the root differential hyperplanes.
+where Bsupset T is a Borel subgroup and b=b=Lie B. Its
+source is closed in (G/B)timesg,g, so the morphism is proper.
+Choose cinfrakt outside the root differential hyperplanes.
 At [1,c], the differential of the incidence morphism is onto (the
 negative-root tangent directions map isomorphically to the negative root
-spaces, and mathfrak bfrakb supplies the rest), so the morphism is dominant,
+spaces, and b supplies the rest), so the morphism is dominant,
 hence surjective.
 
 For the semisimple-conjugation statement: write a semisimple element of
-mathfrak bfrakb as c+u with cinmathfrak tfrakt and u in the positive
+b as c+u with cinfrakt and u in the positive
 root Lie algebra. Conjugate by positive-root groups in increasing height.
 If dalpha(c)ne0, conjugation by the alpha-root group changes the
 alpha coefficient by -z,dalpha(c) and only touch higher-height
@@ -739,19 +739,19 @@ semisimple, u_0 is nilpotent, and their sum is semisimple, so the
 
 
 Choose a closed faithful representation G↪GL(V).
-Its differential realises mathfrak gfrakg as a restricted Lie subalgebra of
+Its differential realises g as a restricted Lie subalgebra of
 End(V), with x|-> x^{[p]} equal to matrix pth power. We use
 the ordinary matrix Jordan decomposition; the two lemmas below prove
-its two parts lie in mathfrak gfrakg in each regime; compare
+its two parts lie in g in each regime; compare
 cite[S4.4, Thm.~4.4.20]{Springer}.
 
 
-Suppose a_0=s+ninmathfrak gfrakg where s,ninmathfrak g,frakg, s is
+Suppose a_0=s+ninfrakg where s,ning,g, s is
 semisimple, n is nilpotent in a faithful representation, and
 [s,n]=0. Suppose also chi(a_0)nechi(0). After conjugation, s is
-a nonzero element of mathfrak t,frakt, M=Z_G(s) is a Levi subgroup,
-a_0inmathfrak m,frakm, and ad_{a_0} is invertible on
-mathfrak g/frakg/mathfrak m.frakm. Moreover for every lambdacolonG_m
+a nonzero element of t,t, M=Z_G(s) is a Levi subgroup,
+a_0inm,m, and ad_{a_0} is invertible on
+g/g/m.m. Moreover for every lambdacolonG_m
 to Z(M),
 
  kappa(a_0,dlambda)=kappa(s,dlambda).
@@ -759,10 +759,10 @@ to Z(M),
 
 
 Proof. 
-The root-pairing lemma conjugates s into mathfrak t.frakt. If s=0 then
-n=a_0 is nilpotent. The projection mathfrak bfrakbtomathfrak tfrakt preserves
+The root-pairing lemma conjugates s into t.t. If s=0 then
+n=a_0 is nilpotent. The projection  tfrakt preserves
 restricted p-operations; since n is killed by some iterated
-p-operation and that operation is injective on mathfrak t,frakt, its
+p-operation and that operation is injective on t,t, its
 toral projection vanishes. A strictly dominant cocharacter contracts
 the positive-root Lie algebra to zero. Therefore every invariant
 polynomial has the same value on n as on 0, contradiction with
@@ -770,7 +770,7 @@ chi(a_0)nechi(0). Thus sne0.
 
 By Hypothesis~hyp:common(iii), M=Z_G(s) is a Levi subgroup. Its
 Lie algebra is kerad_s, which contains a_0. On
-mathfrak g/frakg/mathfrak mfrakm the operator ad_s is diagonalisable with
+g/g/m the operator ad_s is diagonalisable with
 nonzero eigenvalues, while ad_n is nilpotent (indeed
 ad_n^{p^e}=ad_{n^{[p^e]}}=0 for large e) and commutes with
 ad_s. Therefore on each ad_s-eigenspace the sum
@@ -779,11 +779,11 @@ ad_{a_0}=ad_s+ad_n is invertible.
 For (eq:nilpotent-orthogonality): conjugate n inside M into
 the Lie algebra of a Borel of M containing T; the incidence
 argument of the previous lemma applies to M too, and its roots have
-nonzero differentials and kappa is nondegenerate on mathfrak mfrakm by
+nonzero differentials and kappa is nondegenerate on m by
 the same weight decomposition. Restricted nilpotence again kills the
 toral projection, so the conjugated n lies in the positive-root Lie
-algebra of M. That Lie algebra is orthogonal to mathfrak t,frakt, so
-kappa(n,dlambda)=0 for dlambdainmathfrak tfrakt fixed by M; by
+algebra of M. That Lie algebra is orthogonal to t,t, so
+kappa(n,dlambda)=0 for dlambdainfrakt fixed by M; by
 invariance the same holds before conjugation.
 
 
@@ -796,7 +796,7 @@ diverge; everything else is common.
 
 Assume Hypothesis~hyp:A, so k=Fbar_q. Two facts
 are needed. First, the Jordan parts s,n of a_0 constructed in
-Lemma~lie:constant-parts actually lie in mathfrak g;frakg; they start
+Lemma~lie:constant-parts actually lie in g;g; they start
 life only in an ambient End(V). Second, some lambdain
 X_*(Z(M)) has kappa(s,dlambda)ne0. Under Hypothesis~hyp:A,
 both are integral.
@@ -805,9 +805,9 @@ The Jordan parts first. In a faithful representation write a_0=s+n for
 the matrix Jordan decomposition. The eigenvalues of s lie in a finite
 field F_{p^h}; choose e divisible by h and large enough
 to have n^{p^e}=0. Then a_0^{[p^e]}=a_0^{p^e}=s^{p^e}+n^{p^e}=s^{p^e}
-so s=a_0^{[p^e]}, hence sinmathfrak gfrakg as the p^eth restricted
-power of an element of mathfrak g.frakg. Consequently ninmathfrak gfrakg too.
-Lemma~lie:constant-parts then supplies a nonzero sinmathfrak t,frakt,
+so s=a_0^{[p^e]}, hence sinfrakg as the p^eth restricted
+power of an element of g.g. Consequently ninfrakg too.
+Lemma~lie:constant-parts then supplies a nonzero sint,t,
 M=Z_G(s) and the quotient invertibility.
 
 For the cocharacter, we use the following lattice statement. It uses
@@ -819,28 +819,28 @@ Cartan determinant is a p-adic unit. Moreover, for every Levi
 Msupset T,
 
  span_k{dlambda: lambdain X_*(Z(M))}
- =mathfrak z(frakz(mathfrak m),frakm),
+ =z(z(m),m),
 
-where the dlambdainmathfrak tfrakt are differentials of actual
+where the dlambdainfrakt are differentials of actual
 central cocharacters of Z(M).
 
 
 Proof. 
 Fix a positive system and let I={alpha_1,...,alpha_r} be the
 simple roots of M. Set
-J_I=span_k{dalpha_i^vee}subsetmathfrak t.frakt. A
-central element of mathfrak mfrakm has no root-space component because all
+J_I=span_k{dalpha_i^vee}subsett.t. A
+central element of m has no root-space component because all
 root differentials are nonzero (Lemma~lie:root-pairing), so
 (eq:root-coroot-pairing) gives
 [
- mathfrak z(frakz(mathfrak m)=∩_ifrakm)=∩_iker dalpha_i=J_I^perp.
+ z(z(m)=∩_ifrakm)=∩_iker dalpha_i=J_I^perp.
 ]
 The form on this center is nondegenerate by hypothesis, and on
-mathfrak tfrakt by Lemma~lie:root-pairing; nondegeneracy on J_I
+t by Lemma~lie:root-pairing; nondegeneracy on J_I
 follows.
 
-Define integral maps UcolonΛtoZ^r,
-y|->(<alpha_i,y>)_i, and VcolonZ^r
+Define integral maps U:ΛtoZ^r,
+y|->(<alpha_i,y>)_i, and V:Z^r
 Λ, e_j|->alpha_j^vee. Their composite is the Cartan matrix
 C_I=UV. Reduction to k is denoted by a subscript; by
 (eq:root-coroot-pairing) we have V_k^*kappa=D U_k for D a
@@ -884,16 +884,16 @@ Pi_I=d_Iid_Λ-Vadj(C_I)UinEnd(Λ). Then
 UPi_I=0; the lattice ker U is exactly the set of integral
 cocharacters into Z(M). For zinker U_k, Pi_{I,k}(z)=d_I z; since
 d_I is invertible in k, the differentials of ker U span
-ker U_k=mathfrak z(frakz(mathfrak m),frakm), giving
+ker U_k=z(z(m),m), giving
 (eq:central-span). For r=0 take d_I=1 and Pi_I=id; the
 same proof covers tori.
 
 
 Now return to the regime-A story. Under Hypothesis~hyp:A, the
-Jordan parts s,ninmathfrak gfrakg have been constructed. Lemma~lie:constant-parts
-gives sinmathfrak z(frakz(mathfrak m)∖frakm)∖{0} and the quotient
-invertibility. Since kappa is nondegenerate on mathfrak z(frakz(mathfrak
-m)frakm) by Hypothesis~hyp:common(i),
+Jordan parts s,ninfrakg have been constructed. Lemma~lie:constant-parts
+gives sinz(z(m)∖m)∖{0} and the quotient
+invertibility. Since kappa is nondegenerate on z(z(mathfrak
+m)m) by Hypothesis~hyp:common(i),
 Lemma~lie:lattice produces an actual lambdain X_*(Z(M))
 with kappa(s,dlambda)ne0. Orthogonality
 (eq:nilpotent-orthogonality) gives the required
@@ -911,7 +911,7 @@ In the faithful representation write a_0=s+n and choose e with
 n^{p^e}=0, so a_0^{p^e}=s^{p^e}. Let V_0subset k be the
 finite-dimensional F_p-spanbbF_p-span of the eigenvalues of s, and let
 V_e={v^{p^e}:vin V_0}. The inverse of Frobenius V_eto V_0 is
-F_p-linear.bbF_p-linear. If V_0=0 then s=0 and a_0=ninmathfrak g,frakg,
+F_p-linear.F_p-linear. If V_0=0 then s=0 and a_0=ning,g,
 contradicting chi(a_0)nechi(0)
 (Lemma~lie:constant-parts). Hence r=dim_{F_p} and
 we may assume V_0ne0.
@@ -924,35 +924,35 @@ le p^{r-1} would vanish on all p^r elements of V_e, contradicting
 Q(z)=sum_{j=0}^{r-1}c_j z^{p^j} agreeing with inverse-Frobenius on
 V_e. Applying it to the diagonalisable matrix s^{p^e} gives
 [
- s=Q(a_0^{p^e})=sum_{j=0}^{r-1}c_j a_0^{[p^{e+j}]}inmathfrak g.frakg.
+ s=Q(a_0^{p^e})=sum_{j=0}^{r-1}c_j a_0^{[p^{e+j}]}ing.g.
 ]
-Hence ninmathfrak gfrakg as well. Lemma~lie:constant-parts now
-supplies sinmathfrak t∖frakt∖{0}, M=Z_G(s) and the quotient
+Hence ninfrakg as well. Lemma~lie:constant-parts now
+supplies sint∖t∖{0}, M=Z_G(s) and the quotient
 invertibility.
 
 To find lambda: let W_M=N_M(T)/T be the Weyl group of M and set
 h_M=|W_M|. Since W_Msubset W_G, Hypothesis~hyp:B implies
 h_M is invertible in k. The averaging operator
 [
- e_M=frac1{h_M}sum_{win W_M}wcolonmathfrak tfrakttomathfrak tfrakt
+ e_M=frac1{h_M}sum_{win W_M}w: tfrakt
 ]
 is a self-adjoint projection for kappa, so kappa is nondegenerate
-on mathfrak t^frakt^{W_M}=im e_M. That subspace contains s because M
-centralises s. In fact mathfrak t^frakt^{W_M}=mathfrak z(frakz(mathfrak m):frakm):
+on t^t^{W_M}=im e_M. That subspace contains s because M
+centralises s. In fact t^t^{W_M}=z(z(m):m):
 the reflection formula is s_alpha(h)=h-dalpha(h)dalpha^vee
 and dalpha^veene0 by Lemma~lie:root-pairing. Thus nondegeneracy
-on mathfrak z(frakz(mathfrak m)frakm) again follows from kappa alone, without
+on z(z(m)m) again follows from kappa alone, without
 needing Hypothesis~hyp:common(i)'s second clause.
 
 For an integral cocharacter, choose a basis mu_1,...,mu_{N_T} of
 Λ and set nu_i=sum_{win W_M}wmu_iinΛ^{W_M}. Their
-differentials span mathfrak t^frakt^{W_M} because dnu_i=h_M e_M(dmu_i).
+differentials span t^t^{W_M} because dnu_i=h_M e_M(dmu_i).
 The integral reflection formula s_alpha(lambda)=lambda-
 <alpha,lambda>alpha^vee and torsion-freeness of
 Λ show that Λ^{W_M}=X_*(Z(M)): a W_M-invariant
 cocharacter has zero pairing with every root of M, and conversely. No
 division of a cocharacter by h_M is made. Nondegeneracy of kappa on
-mathfrak t^frakt^{W_M} now gives an integral lambdain X_*(Z(M)) with
+t^t^{W_M} now gives an integral lambdain X_*(Z(M)) with
 kappa(s,dlambda)ne0, and (eq:nilpotent-orthogonality) gives
 kappa(a_0,dlambda)ne0.
 
@@ -963,36 +963,36 @@ pnmid|W_G| is specific to Hypothesis~hyp:B.
 
 
 Both regimes have now produced the same constant-term data: a Levi M,
-a central cocharacter lambda, and a_0=a(0)inmathfrak mfrakm with
+a central cocharacter lambda, and a_0=a(0)infrakm with
 kappa(a_0,dlambda)ne0 and ad_{a_0} invertible on
-mathfrak g/frakg/mathfrak m.frakm. The remaining step moves the entire formal
-arc a(t) into mathfrak m[[t]].frakm[[t]].
+g/g/m.m. The remaining step moves the entire formal
+arc a(t) into m[[t]].m[[t]].
 
 
 Let G be a connected reductive group over an algebraically closed
-field k, Msubset G a Levi subgroup, a(t)inmathfrak g[[t]]frakg[[t]] with
-a_0=a(0)inmathfrak m.frakm. If ad_{a_0} is invertible on
-mathfrak g/frakg/mathfrak m,frakm, then there is g(t)in G(k[[t]]) with
-g(0)=1 and Ad(g(t))a(t)inmathfrak m[[t]].frakm[[t]].
+field k, Msubset G a Levi subgroup, a(t)ing[[t]]g[[t]] with
+a_0=a(0)inm.m. If ad_{a_0} is invertible on
+g/g/m,m, then there is g(t)in G(k[[t]]) with
+g(0)=1 and Ad(g(t))a(t)inm[[t]].m[[t]].
 
 
 Proof. 
 Proceed by successive approximation. Suppose the arc already lies in
-mathfrak m[[t]]frakm[[t]] modulo t^r, rge1, and let c_rin
-mathfrak g/frakg/mathfrak mfrakm be the class of its t^r-coefficient. Choose
-Dinmathfrak gfrakg with c_r+[D,a_0]=0 in the quotient (possible by
+m[[t]]m[[t]] modulo t^r, rge1, and let c_rin
+g/g/m be the class of its t^r-coefficient. Choose
+Dinfrakg with c_r+[D,a_0]=0 in the quotient (possible by
 invertibility). Smoothness of G gives g_r(t)in G(k[[t]]) with
 g_r(t)equiv1+t^rDpmod{t^{r+1}}; formally, this is a lift of a point
 in the tangent kernel of the square-zero ideal (t^r)/(t^{r+1}) to
 G(k[t]/(t^{r+1})), obtained by iterating formal smoothness at G. Then
 conjugation by g_r(t) changes the t^r-coefficient by [D,a_0] and
 leaves all lower coefficients fixed, so it places the arc in
-mathfrak m[[t]]frakm[[t]] modulo t^{r+1}.
+m[[t]]m[[t]] modulo t^{r+1}.
 
 Iterating, the products prod g_r(t) converge in the t-adic
 topology because g_requiv1pmod{t^r}. The affine group G then
 produces a single g(t)in G(k[[t]]) with g(0)=1 and
-Ad(g(t))a(t)inmathfrak m[[t]].frakm[[t]]. No exponential series is used.
+Ad(g(t))a(t)inm[[t]].m[[t]]. No exponential series is used.
 
 
 Proof. [Proof of Proposition~lie:input]
@@ -1008,8 +1008,8 @@ construction, so all of (eq:lie-output) holds.
 
 
 Centrality of lambda in M gives Ad(lambda(t))a(t)=a(t), and
-every nonzero lambda-weight space of mathfrak gfrakg injects into
-mathfrak g/frakg/mathfrak m,frakm, so the adjoint action is invertible on the
+every nonzero lambda-weight space of g injects into
+g/g/m,m, so the adjoint action is invertible on the
 truncated nonzero-weight modules used below. We do not need
 Z_G(lambda)=M. If lambda is central in G the Hecke orbit has
 dimension zero, but the nonzero pairing in (eq:lie-output) still
@@ -1070,7 +1070,7 @@ Let V be the irreducible representation of check G corresponding
 to this bound, with the indicated input/output convention. Then the
 generic Hecke functor has the form
 
- mathrm H_V(F)=rmH_V(F)=overlineoverline q_q_{K,*}I_K,
+ H_V(F)=rmH_V(F)=overlineoverline q_q_{K,*}I_K,
 
 with I_K on H_K equal to p_K^*F tensored with a fixed
 invertible constant complex (the perverse intersection-cohomology
@@ -1129,9 +1129,9 @@ element of H^0 of the cotangent-complex fibre. Deformation theory and
 Serre duality cite[Tag~0BS4]{Stacks}, together with kappa, identify
 this space as
 [
- T_P^*B_s=H^1(X,mathfrak g_P)^*=H^0(X,frakg_P)^*=H^0(X,mathfrak g_Pfrakg_P ⊗ omega_X),
+ T_P^*B_s=H^1(X,g_P)^*=H^0(X,g_P)^*=H^0(X,g_P ⊗ omega_X),
 ]
-where mathfrak g_Pfrakg_P is the adjoint bundle. We call such a vector A a
+where g_P is the adjoint bundle. We call such a vector A a
 Higgs field. For a representable smooth map, the cotangent
 pullback is injective and its image is the kernel of passage to the
 relative cotangent; we use this in the form supplied by the cotangent
@@ -1156,7 +1156,7 @@ Moreover, on the fixed-side fibres
 ]
 with fixed-bundle identifications retained, the following hold. The
 restriction of p^*A to the q-relative tangent bundle defines a
-section S_A of Ω_{H_s/Z_s,q}calZ_s,q; the restriction of
+section S_A of Ω_{H_s/Z_s,q}Z_s,q; the restriction of
 o^*A' to the widetilde p-relative tangent bundle defines a section
 S_{A'}. Both sections vanish at h, and both derivatives
 
@@ -1178,8 +1178,8 @@ chi(a(0))nechi(0). Proposition~lie:input then supplies a
 formal change of frame, a Levi M, and a cocharacter
 lambdacolonG_m Z(M) with
 
- a(t)inmathfrak m[[t]],frakm[[t]],
- ad_{a(0)} invertible on mathfrak g/frakg/mathfrak m,frakm,
+ a(t)inm[[t]],m[[t]],
+ ad_{a(0)} invertible on g/g/m,m,
  kappa(a(0),dlambda)ne0.
 
 After constant conjugation we regard lambda as an element of the
@@ -1193,13 +1193,13 @@ cite[SS20.4--20.6]{AGKRRV}. What follows makes both derivative
 assertions explicit.
 
 Tangent spaces. In the input frame, put
-L=mathfrak g[[t]],frakg[[t]], L'=Ad(lambda(t))L, I=Lcap L'. Let
-mathfrak g_jfrakg_j be the weight-j space of lambda; the integers j
+L=g[[t]],g[[t]], L'=Ad(lambda(t))L, I=Lcap L'. Let
+g_j be the weight-j space of lambda; the integers j
 are genuine characters of G_m, not reduced mod p. Then
 
- L'=⊕_j t^jmathfrak g_j[[t]],frakg_j[[t]],
- P_1:=L/I=⊕_{j>0}mathfrak g_jfrakg_j ⊗  k[[t]]/(t^j),
- P_2:=L'/I=⊕_{j<0}mathfrak g_jfrakg_j ⊗  (t^jk[[t]]/k[[t]]).
+ L'=⊕_j t^jg_j[[t]],g_j[[t]],
+ P_1:=L/I=⊕_{j>0}g_j ⊗  k[[t]]/(t^j),
+ P_2:=L'/I=⊕_{j<0}g_j ⊗  (t^jk[[t]]/k[[t]]).
 
 An infinitesimal disc automorphism 1+ε D, Din L, acts on
 the output lattice while fixing (P,x), and its stabiliser tangent is
@@ -1218,10 +1218,10 @@ There is a perfect pairing
  beta: P_1times P_2to k,
  beta(D,C)=Res_{t=0}kappa(D,C),dt.
 
-Indeed, L is its own residue-annihilator on mathfrak g((t)),frakg((t)), and
+Indeed, L is its own residue-annihilator on g((t)),g((t)), and
 invariance of kappa gives the same statement for L'. So beta
-descends to the quotients. Concretely, kappa pairs mathfrak g_jfrakg_j
-perfectly with mathfrak g_frakg_{-j} and the coefficient of t^a in the
+descends to the quotients. Concretely, kappa pairs g_j
+perfectly with g_{-j} and the coefficient of t^a in the
 first space, for 0le a<j, pairs with the coefficient of t^{-a-1}
 in the second. This is perfect, including when j is divisible by p.
 
@@ -1236,7 +1236,7 @@ dq_h^*theta for a unique covector theta.
 The bundle component of theta is A'. Indeed, choose x'ne x and
 vary both bundle transitions only at x', leaving the modification at
 x unchanged. Principal parts at x' surject to
-H^1(X,mathfrak g_frakg_{P'}) (Serre vanishing for Ngg0), and the
+H^1(X,g_{P'}) (Serre vanishing for Ngg0), and the
 pairings agree on these directions since A=A' near x'. Write
 theta=(A',xi).
 
@@ -1272,7 +1272,7 @@ preserves L,L',I, the formula is independent of representatives.
 For invertibility: ad(a(t)) preserves every weight summand in
 (hecke:weight-quotients). Because lambda is central in M,
 each nonzero weight space injects as an invariant summand into
-mathfrak g/frakg/mathfrak m;frakm; its endomorphism from ad(a(0)) is therefore
+g/g/m;m; its endomorphism from ad(a(0)) is therefore
 invertible by (hecke:lie-data). The corresponding k[[t]]-linear
 block has unit determinant, so ad(a(t)) is invertible on every
 truncated summand of P_1 and P_2. Combined with
@@ -1366,7 +1366,7 @@ after restricting to the open orbit.
 Step 1: a proper fibre with vanishing cohomology.: 
 Write D=V(f). Take two copies H_1,H_2 of H and form
 [
- W_0=H_2times_{p_2,B,b}calB,bU,
+ W_0=H_2times_{p_2,B,b}B,bU,
  W=H_2times_{B} W_0.
 ]
 Both map to Z= BcalBtimes_SX by
@@ -1380,7 +1380,7 @@ criterion therefore makes WtoB (via the output bundle)
 smooth near w. The arbitrary-leg assertion of
 Lemma~spec:vanishing gives
 
- (Psi_W(q_{2,K}^*mathrm H_V(F)))_w=0.rmH_V(F)))_w=0.
+ (Psi_W(q_{2,K}^*H_V(F)))_w=0.rmH_V(F)))_w=0.
 
 Note: smoothness is used only for the projection to B, matching
 the hypothesis of Lemma~spec:vanishing.
@@ -1414,7 +1414,7 @@ z=(h,h,u)in P_D(k).
 
 Let Ibar_K be the pullback of the Satake integrand
 to (overlineoverline P_D)_K.P_D)_K. Proper base change on the generic fibre gives
-Rpi_{K,*}Ibar_K~= q_{2,K}^*mathrm H_V(F).rmH_V(F).
+Rpi_{K,*}Ibar_K~= q_{2,K}^*H_V(F).rmH_V(F).
 Apply proper compatibility of specialisation, then proper base change
 at w, to (eq:two-hecke-zero). With
 [
@@ -1480,7 +1480,7 @@ is a map r: Pto U with an isomorphism
  br~= a, r|_Delta=c|_Delta,
 
 whose restriction to Delta is the specified isomorphism. For this,
-form the smooth space E_P=Ptimes_{a,B,b}calB,bUto P. The
+form the smooth space E_P=Ptimes_{a,B,b}B,bUto P. The
 diagonal data give a section over Delta. First take an 'etale
 scheme neighbourhood in P. Since B has affine diagonal
 (Section~sec:specialization), E_Pto Ptimes_S U is affine, so
@@ -1632,7 +1632,7 @@ smaller quadric). The resulting filtration by affine cells has one cell
 in every dimension 0,...,n, and a second middle-dimensional cell
 when n is even. Initial cases are a conic (congP^1) and
 the split zero-dimensional quadric (two points). Localization plus
-RGamma_c(A^r,E)=E(-r)[-2r]bbA^r,E)=E(-r)[-2r] gives the cohomology. On a smooth
+RGamma_c(A^r,E)=E(-r)[-2r]A^r,E)=E(-r)[-2r] gives the cohomology. On a smooth
 quadric of positive dimension the top hyperplane power has degree 2,
 hence is nonzero in E; all lower powers are nonzero as well.
 
@@ -2035,7 +2035,7 @@ Step 3: choose transverse equations.
 Let V=T_u^*U_s, so dim V=N. If d>0, a general d-plane
 Lsubset V satisfies LcapΛ_u⊆{0}: from
 (eq:support-cone-fiber),
-dimP(Λ_u)bbP(Λ_u)le N-d-1, and the incidence of d-planes
+dimP(Λ_u)P(Λ_u)le N-d-1, and the incidence of d-planes
 meeting this projective set is a proper closed subset of Gr(d,V). The
 condition that Lto T_u^*T be an isomorphism is another nonempty open
 condition; two nonempty opens meet because the Grassmannian is
@@ -2205,7 +2205,7 @@ determine geometric q-Frobenius, whose pullback action on QLisse(X)
 induces an automorphism Frob of Y. Define the derived fixed-point
 stack
 [
- Y^{arithm}:=Y^{Frob}=Yunderset{Ytimes_EY}{overset{mathrm R}rmRtimes}Y,
+ Y^{arithm}:=Y^{Frob}=Yunderset{Ytimes_EY}{overset{R}rmRtimes}Y,
 ]
 where the two maps to Ytimes_EY are the diagonal and the graph of
 Frob. An E-point of Y^{arithm} is a geometric local
@@ -2216,7 +2216,7 @@ and cite[S1.5.1]{GR}.
 For the finite-field datum of Hypothesis~hyp:A there is a
 canonical isomorphism
 [
- Funct_c(Bun_{G_0}(F_q),E)bbF_q),E)
+ Funct_c(Bun_{G_0}(F_q),E)F_q),E)
  ~=Gamma^{IndCoh}(Y^{arithm},omega_{Y^{arithm}}).
 ]
 The left side is H^0 of compactly supported E-valued functions on
@@ -2231,7 +2231,7 @@ calculation of Beraldo--Lin--Reeves
 cite[SS6.4.12--6.4.13]{BeraldoLinReeves} enters the known primed
 formula cite[SS1.5.3--1.5.5,~Cor.~1.5.6]{GR}:
 [
- Funct_c(Bun_{G_0}(F_q),E)bbF_q),E)
+ Funct_c(Bun_{G_0}(F_q),E)F_q),E)
  ~=Gamma^{IndCoh}((Y')^{Frob},omega_{(Y')^{Frob}}).
 ]
 Its automorphic trace input is cite[Thm.~0.2.6]{AGKRRVTrace} with
@@ -2259,7 +2259,7 @@ categorical input is simple.
 Let A_0 be a compactly generated, presentable, stable,
 E-linear symmetric monoidal category whose compact objects are
 dualizable, and suppose its tensor product preserves colimits
-separately. Let acolonA_0 be a continuous
+separately. Let a:A_0 be a continuous
 E-linear symmetric monoidal functor, and give Vect_E the
 A_0-actioncalA_0-action through a. Then the right adjoint R_a of a
 is continuous, carries a coherent A_0-linearcalA_0-linear structure, and
@@ -2289,7 +2289,7 @@ coherences before invertibility is imposed; invertibility then gives a
 coherent A_0-linearcalA_0-linear structure on R_a.
 
 Finally R_a(E)ne0 because
-RHom_{A_0} 1,R_a(E))bf1,R_a(E))
+RHom_{A_0} 1,R_a(E))1,R_a(E))
 ~=RHom_E(E,E)~= E; this last argument does not require a
 compact unit.
 
@@ -2300,7 +2300,7 @@ cite[S7.9,~Cor.~C.4.4]{AGKRRV}. On a disjoint union of components the
 compact generators have finite component support. We use the continuous,
 fully faithful A-linearcalA-linear functor
 [
- jmath_YcolonQCoh(Y)xrightarrow[sim]{Υ_Y}
+ jmath_Y:QCoh(Y)xrightarrow[sim]{Υ_Y}
  IndCoh_{{0}}(Y)↪IndCoh_{Nilp}(Y),
 ]
 where Υ_Y(Q)=Q ⊗ omega_Y
@@ -2312,7 +2312,7 @@ VinRep(check G)^{ ⊗  I}, let H_I(V,-) denote the multi-leg
 Hecke functor and let
 E_V^I AcalA ⊗ Q_I be the universal
 evaluation object. For a parameter sigmacolonSpec Eto Y define
-sigma^I(V):=(sigma^* ⊗ id)(E_V^I) Q_I.calQ_I.
+sigma^I(V):=(sigma^* ⊗ id)(E_V^I) Q_I.Q_I.
 
 
 A Hecke eigenobject of eigenvalue sigma with coherent structure is an
@@ -2341,7 +2341,7 @@ The composite
  xrightarrow{jmath_Y}D
  xrightarrow{(L_G^{restr})^{-1}}C_s
 ]
-is continuous A-linear,calA-linear, where Vect_sigma denotes
+is continuous A-linear,A-linear, where Vect_sigma denotes
 Vect_E with the action through sigma^*. Set
 M_sigma=Phi_sigma(E). It is nonzero by the lemma, full faithfulness
 of jmath_Y, and the Langlands equivalence. If sigma lies in Z,
@@ -2427,7 +2427,7 @@ Extension of coefficients induces equivalences
  
  Vect_E ⊗ _{E_0}QLisse_{E_0}(X)&~=QLisse_E(X),
  
- Vect_E ⊗ _{E_0}C_0&calC_0&~=C_E.
+ Vect_E ⊗ _{E_0}C_0&C_0&~=C_E.
  
 
 The first extends to locally finite-type algebraic stacks by smooth
@@ -2726,7 +2726,7 @@ as in Theorem~thm:rational-support. In addition to the rational
 hypotheses, assume a canonical primed trace isomorphism over E_0 is
 given,
 [
- Funct_c(Bun_{G_0}(F_q),E_0)bbF_q),E_0)
+ Funct_c(Bun_{G_0}(F_q),E_0)F_q),E_0)
  xrightarrow{ sim }
  Gamma^{IndCoh}((Y'_0)^{Frob},omega_{(Y'_0)^{Frob}}),
 ]
@@ -2860,7 +2860,7 @@ arXiv:2410.10692v2, 21 October 2024.
 bibitem[Conrad]{Conrad}
 B.~Conrad,
 Reductive group schemes,
-available at https://math.stanford.edu/~conrad/papers/luminysga3smf.pdf, 2014.
+available at ./pdf/unknown-nd-available-at.pdf, 2014.
 
 bibitem[Milne]{Milne}
 J.~S.~Milne,
@@ -2871,7 +2871,7 @@ bibitem[MilneEtale]{MilneEtale}
 J.~S.~Milne,
 Lectures on 'Etale Cohomology,
 version 2.21, 22 March 2013;
-https://www.jmilne.org/math/CourseNotes/LEC.pdf.
+./pdf/unknown-nd.pdf.
 
 bibitem[FrenkelGaitsgoryVilonen]{FrenkelGaitsgoryVilonen}
 E.~Frenkel, D.~Gaitsgory, K.~Vilonen,