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Equivariant and twisted D-modules

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

A symmetry of a differential equation can mean two different things. The group may transport the equation compatibly, or that transport may itself be horizontal in the group direction. The second condition is what allows the equation to live on a quotient stack. A twist changes another part of the data: it changes how local differential operators are identified when a line bundle changes frame.

We work over an algebraically closed field $k$ of characteristic zero. Varieties are smooth, and left D-modules are quasi-coherent over the structure sheaf. Groups are smooth algebraic groups of finite type; the Harish-Chandra and derived quotient constructions below use affine groups. We use ordinary, unshifted smooth pullback for the abelian descent category. For a smooth map of relative dimension $r$ it equals $f^![-r]$ in the conventions of Inverse images. Derived descent retains the homotopies as well as these transition maps.

The background is group actions and rational representations, faithfully flat descent for quasi-coherent modules, and the quotient stack defined by its torsors. The Weyl algebra computations use The Riemann-Hilbert correspondence.

1. When equivariance is horizontal

Let $G$ act on $X$. Its action on functions is \[ (g\cdot f)(x)=f(g^{-1}x). \] Write $\mu:\mathfrak g\to\Gamma(X,T_X)$ for the differentiated action on functions. This is a Lie algebra homomorphism. With this convention $\mu(\xi)$ is the negative of the vector field obtained by differentiating $\exp(t\xi)x$. In particular, for scaling on $\mathbb A^1$, \[ \mu(1)=-x\partial_x. \tag{1.1} \]

A weakly equivariant D-module is an algebraic $G$-linearization of its quasi-coherent $\mathcal O_X$-module, compatible with the transported D-action. Thus \[ g(Pm)=(gP)(gm). \] The linearization differentiates to an action $d\rho$ of $\mathfrak g$ on sections. Both $d\rho(\xi)$ and $\mu(\xi)$ acting through $\mathcal D_X$ satisfy the same Leibniz rule. Weak compatibility also gives \[ [d\rho(\xi),P]=[\mu(\xi),P]. \] Consequently their difference \[ \xi^\natural=d\rho(\xi)-\mu(\xi) \tag{1.2} \] is D-linear. The differences form a Lie algebra action: expand their bracket, use the preceding identity with $P=\mu(\eta)$, and obtain $[\xi^\natural,\eta^\natural]=[\xi,\eta]^\natural$.

The module is strongly equivariant if every $\xi^\natural$ is zero. We will use “equivariant” in this strong sense. This is the condition in Beilinson–Drinfeld, §7.6.10.

For example $\mathcal O_G$, with its usual differentiation and left translation, is strongly equivariant. On a point, however, the D-action of $\mathfrak g$ is zero. A weakly equivariant D-module on a point is any rational $G$-representation; it is strong precisely when its differentiated representation is zero.

Theorem 1.1. If $G$ is connected, forgetting the strong linearization is fully faithful.

Proof. First observe a useful relative horizontality fact. For any quasi-coherent module $M$ on $X$, a map between two pullbacks from $X$ to $G\times X$ which commutes with differentiation in the $G$ direction is determined by its restriction to $\{1\}\times X$.

To check this, take affine opens $U\subset X$, $W\subset G$. The image of any local section is a finite sum $\sum_i f_i\otimes m_i$, with $f_i\in\mathcal O(W)$ and with the $m_i$ linearly independent over $k$. Differentiating in $G$ makes every $df_i$ zero. A regular function with zero differential on a smooth connected variety in characteristic zero is constant: its image in the function field has zero differential, hence is algebraic over $k$, hence belongs to $k$. A connected smooth algebraic group is irreducible, so the constants agree on overlapping opens. This proves the fact without a finite-rank hypothesis on $M$.

Now write $a,p:G\times X\rightrightarrows X$ for action and projection. A strong linearization is a D-linear isomorphism between their unshifted smooth pullbacks. Indeed weak compatibility is D-linearity in the $X$ directions, and (1.2) is exactly the remaining condition in the $G$ directions. Those directions are generated by translated elements of $\mathfrak g$.

Given a D-linear map $u:M\to N$, compare the two maps obtained by pulling $u$ back and using the two linearizations. After using one linearization to identify action and projection pullbacks, their difference is horizontal along $G$. Its restriction at the identity is zero. The relative fact shows that it vanishes. Thus $u$ is equivariant. Faithfulness is immediate. Applying the same argument to the identity of $M$ shows that two strong linearizations on $M$ coincide. $\square$

Existence remains a condition. A module need not admit such a linearization. Connectedness also matters: for a finite group on a point, any representation is strong, and different representations can have the same underlying vector space.

2. The quotient category

Put $\mathcal Y=[X/G]$. Its atlas $q:X\to\mathcal Y$ has groupoid \[ X\times_{\mathcal Y}X=G\times X,\qquad X\times_{\mathcal Y}X\times_{\mathcal Y}X=G\times G\times X. \] A D-module on $\mathcal Y$ in the abelian sense means cartesian D-module data on smooth charts, using unshifted smooth pullback. In particular its datum on this atlas is a D-module $M$ on $X$ with a D-linear isomorphism \[ p^\dagger M\simeq a^\dagger M \tag{2.1} \] satisfying the identity and multiplication cocycles.

Theorem 2.1. There is an equivalence \[ \operatorname{Mod}(\mathcal D_{[X/G]}) \simeq\operatorname{Mod}_G(\mathcal D_X). \tag{2.2} \] If $X\to Y$ is a principal $G$-bundle with smooth variety quotient $Y$, this category is $\operatorname{Mod}(\mathcal D_Y)$.

Proof. Forgetting derivatives in the $G$ direction in (2.1) gives an $\mathcal O$-linearization with the group cocycle. D-linearity in the $X$ directions gives weak compatibility. D-linearity in the $G$ directions, differentiated at the identity, gives $d\rho=\mu$. Conversely these two conditions give D-linearity in every direction on $G\times X$, so a strong linearization supplies (2.1).

This gives inverse functors on objects and morphisms for the displayed atlas. To see that it really defines data on all charts, refine any two smooth charts by their fiber product. Quasi-coherent descent glues the modules and maps. Derivations descend because the local actions agree after pullback; their Leibniz rules and bracket relations can be checked on a faithfully flat cover. Smooth maps are locally factorizable into an étale map and a projection, where the horizontal pullback action is the usual one. Thus atlas refinements produce the same cartesian category.

For a principal bundle $[X/G]\simeq Y$: a map to the quotient stack is a $G$-torsor together with an equivariant map to $X$, which is precisely the pullback of $X\to Y$. Equivalently, apply the same effective descent argument directly to $X\to Y$. $\square$

This does not construct the stack category as modules over an ordinary sheaf of rings obtained by descending $\mathcal D_X$. Smooth pullback of a ring of differential operators is not the ring of differential operators upstairs. Beilinson–Drinfeld, §§1.1.3–1.1.5, distinguishes these constructions; its additional “good stack” hypothesis concerns the particular ring of operators it constructs. The cartesian D-module category (2.2) does not require that hypothesis.

There is also a derived distinction. The derived D-module category of $\mathcal Y$ uses homotopy-coherent smooth descent. It need not equal the derived category of the abelian category in (2.2). Section 6 gives a concrete difference.

3. Twisting operators by a line bundle

Let $L$ be a line bundle and $\lambda\in k$. Choose local frames $e_i$ and write $e_j=g_{ij}e_i$. On overlaps identify copies of $\mathcal D_X$ by \[ f\longmapsto f,\qquad \xi\longmapsto \xi+\lambda\,g_{ij}^{-1}\xi(g_{ij}). \tag{3.1} \] Here $\xi$ is a vector field. There is no chosen logarithm; $d\log g$ means $g^{-1}dg$.

Theorem 3.1. These identifications define a sheaf of filtered rings $\mathcal D_L^\lambda$ with \[ \operatorname{gr}\mathcal D_L^\lambda\simeq \operatorname{Sym}_{\mathcal O_X}T_X. \] For $n\in\mathbb Z$, \[ \mathcal D_L^n\simeq \operatorname{Diff}(L^{\otimes n},L^{\otimes n}) \simeq L^{\otimes n}\otimes_{\mathcal O_X}\mathcal D_X \otimes_{\mathcal O_X}L^{\otimes(-n)}, \tag{3.2} \] with the composition multiplication transported to the last expression.

Proof. Adding a function to a vector field preserves its commutator with functions. To check the bracket of two vector fields, put $\alpha=\lambda\,d\log g$. Then \[ [\xi+\alpha(\xi),\eta+\alpha(\eta)] =[\xi,\eta]+\xi\alpha(\eta)-\eta\alpha(\xi) =[\xi,\eta]+\alpha([\xi,\eta]), \] since $d\alpha=0$. Thus (3.1) preserves all defining relations of $\mathcal D_X$. Its inverse uses $g^{-1}$. On triple overlaps $g_{ik}=g_{ij}g_{jk}$ and logarithmic differentials add, proving the cocycle. The maps preserve order and fix principal symbols. PBW therefore identifies the associated graded ring as claimed.

For an integer $n$, a section $s=f_i e_i^n=f_j e_j^n$ satisfies $f_i=g_{ij}^n f_j$. An operator $P_i$ on its coefficients becomes $P_j=g_{ij}^{-n}P_i g_{ij}^n$. On vector fields this is exactly (3.1) with $\lambda=n$. This proves (3.2), including negative $n$. $\square$

A $\lambda$-twisted D-module is a module over this ring, quasi-coherent over $\mathcal O_X$. For integer $n$, tensoring with $L^n$ gives an equivalence from ordinary D-modules to $\mathcal D_L^n$-modules; its inverse tensors with $L^{-n}$. This is Morita equivalence, even when the line bundle has no flat connection.

The punctured line bundle explains arbitrary $\lambda$. Let $\pi:L^\times\to X$, and in a frame write a nonzero vector as $t_i e_i$. Its vertical Euler field is $E=t_i\partial_{t_i}$. Fiber-scaling invariant operators locally form \[ \bigl(\pi_*\mathcal D_{L^\times}\bigr)^{\mathbb G_m} \simeq\mathcal D_{U_i}[E]. \] Indeed a weight-zero fiber monomial is $t^b\partial_t^b$, a falling-factorial polynomial in $E$; Laurent coefficients allow the same reduction for every invariant operator. The element $E$ is central in this invariant ring.

Since $t_i=g_{ij}t_j$, differentiation while holding $t_i$ fixed gives \[ \xi_i=\xi_j-(d\log g_{ij})(\xi)E. \] Consequently \[ \mathcal D_L^\lambda= \bigl(\pi_*\mathcal D_{L^\times}\bigr)^{\mathbb G_m}/(E+\lambda). \tag{3.3} \] The minus sign in the prescribed Euler eigenvalue makes (3.3) agree with (3.2): coefficients of sections of $L^n$ are homogeneous functions of fiber weight $-n$.

One can describe modules by twisted descent on $L^\times$, with this prescribed scalar infinitesimal vertical action. Mere local finiteness of $E$, or allowing a nilpotent part at eigenvalue $-\lambda$, is a larger monodromic category and is not the fixed ring quotient (3.3).

In particular $\mathcal D_{\omega_X}^{1/2}$ exists whether or not $\omega_X$ has a square root. The notation specifies an operator twist, not an assertion that a fractional tensor power exists as a line bundle.

Operators on the projective line

Take $L=\mathcal O(1)$, with coordinate $x$ on one chart and $y=x^{-1}$ on the other. Frames satisfy $e_\infty=x e_0$. Formula (3.1) gives \[ \partial_x=-y^2\partial_y+\lambda y \quad\text{in }\mathcal D_L^\lambda. \tag{3.4} \] The following are global operators: \[ e=\partial_x,\qquad h=-2x\partial_x+\lambda,\qquad f=-x^2\partial_x+\lambda x. \tag{3.5} \] They satisfy \[ [h,e]=2e,\quad [h,f]=-2f,\quad [e,f]=h,\qquad h^2+2h+4fe=\lambda(\lambda+2). \tag{3.6} \] On the other chart they become, respectively, $-y^2\partial_y+\lambda y$, $2y\partial_y-\lambda$, and $\partial_y$, so they are regular everywhere.

For every $\lambda$, these formulas actually identify \[ \Gamma(\mathbb P^1,\mathcal D_{\mathcal O(1)}^\lambda) \simeq U(\mathfrak{sl}_2)/ \bigl(h^2+2h+4fe-\lambda(\lambda+2)\bigr). \tag{3.7} \] Here is a proof that also rules out missing higher-order operators. The order filtration has exact sequences \[ 0\to F_{m-1}\mathcal D_L^\lambda\to F_m\mathcal D_L^\lambda \to\mathcal O(2m)\to0. \] The elementary cohomology calculation on the two affine charts gives $H^1(\mathbb P^1,\mathcal O(2m))=0$ for $m\geq0$. Induction gives $H^1(F_m)=0$, so taking sections is surjective on every principal-symbol quotient. Hence \[ \operatorname{gr}\Gamma(\mathcal D_L^\lambda) =\bigoplus_{m\geq0}H^0(\mathbb P^1,\mathcal O(2m)). \] This is the even Veronese ring $k[s^2,st,t^2]$. The symbols of $e,h,f$ generate it: in homogeneous coordinates they correspond to $s^2,-2st,-t^2$. Their sole relation is $h^2+4fe=0$.

PBW gives the same associated graded ring for the right side of (3.7). Indeed the Casimir is central, and its leading term is the nonzero polynomial $h^2+4fe$. Every element of its principal ideal is its product with an element of $U(\mathfrak{sl}_2)$; leading terms multiply in the polynomial domain. Thus the associated graded ideal is exactly the principal symbol ideal. The filtered map from (3.5) is an isomorphism on associated graded rings, and induction on filtration degree makes it an isomorphism. $\square$

For integer $\lambda=n$, $\mathcal O(n)$ itself is a twisted module. When $n\geq0$, its global sections have basis $1,x,\ldots,x^n$ in the first frame, and (3.5) acts by \[ e(x^j)=j x^{j-1},\quad h(x^j)=(n-2j)x^j,\quad f(x^j)=(n-j)x^{j+1}. \] This finite-dimensional representation is a quotient representation of the infinite-dimensional ring (3.7); that ring is not its endomorphism ring. Negative $n$ gives the same operator construction, although $\mathcal O(n)$ then has no global sections. For $\lambda=1/2$, the ring and its regular left module still exist; there is no line bundle $\mathcal O(1/2)$ on $\mathbb P^1$.

4. Localization from a Harish-Chandra pair

A Harish-Chandra pair $(\mathfrak g,K)$ consists of a Lie algebra, an algebraic group acting on it by Lie automorphisms, and an injective map $\operatorname{Lie}K\to\mathfrak g$ whose differentiated action is the adjoint action. A module $V$ has a $\mathfrak g$-action and a rational $K$-action, compatible with each other, such that \[ d\rho_K(\xi)v=\xi v\quad(\xi\in\operatorname{Lie}K). \tag{4.1} \]

Suppose $S$ is smooth with a compatible $K$-action and a $K$-equivariant Lie map $\mathfrak g\to\Gamma(S,T_S)$ extending $\mu$ on $\operatorname{Lie}K$. For $\mathcal Y=[S/K]$ define \[ \Delta_S(V)=\mathcal D_S\otimes_{U(\mathfrak g)}V. \tag{4.2} \] The right $U(\mathfrak g)$-action on $\mathcal D_S$ is multiplication by its vector fields. Give (4.2) the diagonal $K$-action. It is strong: differentiating on $D\otimes v$ gives \[ [\mu(\xi),D]\otimes v+D\otimes\xi v =\mu(\xi)D\otimes v, \] because $D\mu(\xi)\otimes v=D\otimes\xi v$. Thus Theorem 2.1 descends it to $\Delta(V)$ on $\mathcal Y$.

For a D-module $M$ on $\mathcal Y$, set \[ \Gamma(M)=\Gamma(S,q^\dagger M). \] The vector-field action gives its $\mathfrak g$-action, and the descent linearization gives its $K$-action. Strong equivariance gives (4.1).

Proposition 4.1. Localization is left adjoint to these global sections: \[ \operatorname{Hom}_{\mathcal D_{\mathcal Y}}(\Delta V,M) \simeq \operatorname{Hom}_{(\mathfrak g,K)}(V,\Gamma M). \tag{4.3} \]

Proof. A D-linear map from (4.2) is determined by $1\otimes v$. The balancing relation says exactly that this assignment is $\mathfrak g$-linear. Requiring compatibility with the diagonal $K$-action says exactly that it is $K$-linear. Conversely any such assignment extends by $D\otimes v\mapsto D\phi(v)$. These operations are inverse and natural, proving (4.3). $\square$

Adjunction does not imply equivalence. Transitivity, central characters and positivity will enter Beilinson-Bernstein localization. Central extensions produce twisted versions of (4.2): the central generator is prescribed a scalar, and the corresponding Lie algebroid acts through a TDO. For finite-dimensional algebraic data the same tensor-Hom proof gives the twisted adjunction. The loop-algebra version needs completed enveloping algebras and continuous modules; it cannot be justified by treating the loop algebra as a finite-dimensional pair.

Frenkel, §§7.4–7.5, applies the completed construction to a smooth projective curve and $\mathrm{Bun}_G$. Its local fibers are coinvariants by the stabilizer Lie algebra. At integral level, the TDO acts on an actual determinant/theta line power; at arbitrary level the ring remains defined. This is a cited application here, rather than a proof of uniformization or the construction of its central extension.

5. Scaling and two boundary spaces

For $\mathbb G_m$ acting on $\mathbb A^1$, put $A=k\langle x,\partial\rangle/(\partial x-x\partial-1)$ and $\theta=x\partial$. A rational $\mathbb G_m$-action decomposes into character spaces. By (1.1), strong equivariance is precisely a semisimple integral $\theta$-action: \[ M=\bigoplus_{n\in\mathbb Z}M_n,\qquad \theta|_{M_n}=n,\qquad g|_{M_n}=g^{-n}. \tag{5.1} \] Conversely this formula defines a rational linearization: every vector has finitely many components, $x$ raises $n$ by one and $\partial$ lowers it by one, with exactly the required transformation of operators.

The integer-block reconstruction proved in The Riemann-Hilbert correspondence, Theorem 2.1, now has zero nilpotent parts. Set \[ V_0=M_0,\quad V_1=M_{-1},\qquad a=x:V_1\to V_0,\quad b=\partial:V_0\to V_1. \] Then \[ ab=0,\qquad ba=0. \tag{5.2} \] The first identity is $\theta|_{M_0}=0$; the second is $\partial x|_{M_{-1}}=(\theta+1)|_{M_{-1}}=0$. Both are necessary.

Conversely any two spaces and arrows satisfying (5.2) reconstruct a module. Take $M_n=V_0$ for $n\geq0$ and $M_n=V_1$ for $n\leq-1$. Define the maps from weight $n$ by \[ x_n=\begin{cases}1&n\geq0,\\a&n=-1,\\n+1&n\leq-2,\end{cases} \qquad \partial_n=\begin{cases}n&n\geq1,\\b&n=0,\\1&n\leq-1.\end{cases} \tag{5.3} \] The Weyl relation is immediate on each tail. At weight $0$ its two terms are $1$ and $ab=0$, and at weight $-1$ they are $ba=0$ and $-1$. Thus it holds there too. Moreover $\theta=n$ on every weight. The inverse identifications use the invertible nonexceptional maps, so they identify morphisms with pairs commuting with $a,b$. This proves the classification.

The finite-dimensional diagrams are precisely the coherent modules in this category. One direction follows by choosing bases at weights $0,-1$, which generate the reconstructed module; its Bernstein growth is at most linear, so it is holonomic. Conversely the finite-generation argument of the preceding lesson shows that all weight spaces are finite dimensional. Arbitrary quasi-coherent modules allow arbitrary spaces in (5.2).

The basic examples are

Module $V_0$ $V_1$ $a$ $b$
$\mathcal O_{\mathbb A^1}$ $k$ $0$ $0$ $0$
$\delta_0=A/Ax$ $0$ $k$ $0$ $0$
$j_*\mathcal O_{\mathbb G_m}=k[x,x^{-1}]$ $k$ $k$ $1$ $0$
$j_!\mathcal O_{\mathbb G_m}=A/A\theta$ $k$ $k$ $0$ $1$

The two extensions differ despite having the same restriction to the open orbit. A logarithmic Jordan block with nonzero nilpotent residue, or an Euler connection with nonintegral exponent on $\mathbb G_m$, is monodromic but is not strongly equivariant. For an integral exponent the Laurent Euler connection is isomorphic to $\mathcal O_{\mathbb G_m}$, and its strong linearization exists and is unique.

6. A point quotient remembers higher cohomology

For connected $G$, Theorem 1.1 on a point immediately gives \[ \operatorname{Mod}(\mathcal D_{BG})\simeq\operatorname{Vect}_k. \tag{6.1} \] Existence is the trivial representation. To verify that these are all the objects directly, a strong rational representation has zero derivative. Its orbit maps have zero differential and are constant on connected $G$, by the argument used in Theorem 1.1.

The derived quotient has a different answer. Let $\mathbf1_{BG}$ denote its constant object, allowing the dimension normalization shift appropriate to the chosen tensor unit. Self-Ext is unaffected by that common shift. Homotopy descent along $\mathrm{pt}\to BG$ gives \[ \operatorname{RHom}_{D\text{-}\mathrm{mod}(BG)} (\mathbf1_{BG},\mathbf1_{BG}) \simeq \operatorname{Tot}R\Gamma_{\rm dR}(G^\bullet), \qquad \operatorname{Ext}^*(\mathbf1_{BG},\mathbf1_{BG}) =H^*_{\rm dR}(BG). \tag{6.2} \] Indeed self-Hom of the pulled-back constant D-module on each smooth level is its unshifted de Rham complex, by the Spencer resolution. Composition is multiplication on these complexes. The descent definition makes the compatible maps and their homotopies the totalization, proving (6.2).

Over $\mathbb C$, (6.2) is $H^*(BG^{\rm an},\mathbb C)$, the cohomology of the topological classifying space. Apply the algebraic de Rham comparison stated in The de Rham functor at every level $G^q$. The resulting first-quadrant descent spectral sequences compare term by term; only finitely many terms contribute to each total degree. Over a general $k$, (6.2) means algebraic de Rham cohomology.

For $G=\mathbb G_m$ this algebra can be computed completely: \[ H^*_{\rm dR}(B\mathbb G_m)=k[c],\qquad |c|=2. \tag{6.3} \] The de Rham cohomology of $\mathbb G_m$ is $\Lambda(u)$, $u=d x/x$ in degree one; multiplication pulls $u$ back to $u\otimes1+1\otimes u$. On the nerve, normalized cosimplicial cohomology is the reduced cobar complex of this exterior coalgebra. Its term in nerve degree $q$ is spanned by $u^{\otimes q}$, in de Rham degree $q$. The differential is zero because the reduced coproduct of $u$ is zero. All terms lie on the diagonal of the spectral sequence. A differential $d_r$ changes the two degrees by $(r,1-r)$, and cannot join two diagonal terms for an integer $r\geq1$. Thus it collapses. Shuffle multiplication gives $[u^{\otimes p}][u^{\otimes q}]=\binom{p+q}{p}[u^{\otimes(p+q)}]$: the odd de Rham degree and the nerve suspension cancel in the shuffle signs. Rescaling by factorials, possible in characteristic zero, gives exactly the polynomial algebra (6.3).

Arinkin–Gaitsgory, §11.2, identifies \[ D\text{-}\mathrm{mod}(B\mathbb G_m) \simeq\operatorname{QCoh} (\mathrm{pt}\mathbin{\times^{\mathbf R}_{\mathbb A^1}}\mathrm{pt}). \tag{6.4} \] The functions on this derived intersection are $R=k[\epsilon]/(\epsilon^2)$, $|\epsilon|=-1$, with zero differential. Its augmentation module $k$ corresponds to the constant object, up to normalization. A semifree resolution of $k$ is \[ P=\bigoplus_{n\geq0}R e_n,\quad |e_n|=-2n,\quad d e_0=0,\quad d e_n=\epsilon e_{n-1}\ (n\geq1). \tag{6.5} \] Each negative-degree cycle $\epsilon e_{n-1}$ is the boundary of $e_n$; the only cohomology is $k e_0$ in degree zero. Applying $\operatorname{Hom}_R(-,k)$ gives one copy of $k$ in each even nonnegative degree and zero differential. The degree-two chain map $e_n\mapsto e_{n-1}$, $e_0\mapsto0$, lifts the first generator; all its iterates remain nonzero after augmentation. Its Yoneda algebra is therefore $k[c]$, independently checking (6.3).

The constant object is not compact here. A compact $R$-module is a retract of a finite semifree module; since $R$ is bounded, such a module has bounded self-Ext. Equation (6.5) gives unbounded self-Ext for $k$. The free module $R$ is instead a compact generator. This is precisely the distinction made in the remark following the torus comparison in Arinkin–Gaitsgory, §11.2. Deriving the vector-space heart in (6.1) would give zero positive self-Ext, so it cannot produce this quotient category.

7. The half-twist on the moduli of bundles

Let $C$ be a smooth complete curve and $G$ a connected reductive group. On $\mathrm{Bun}_G(C)$ the adjoint bundle of a principal bundle $\mathcal P$ defines the normalized determinant line \[ \det_{\mathrm{Bun}_G}|_{\mathcal P} =\det R\Gamma(C,\mathfrak g_{\mathcal P}) \otimes\det R\Gamma(C,\mathfrak g_{\mathcal P_{\rm triv}})^{-1}. \tag{7.1} \] The determinant uses the derived cohomology complex: for a vector bundle it is $\det H^0\otimes(\det H^1)^{-1}$. Gaitsgory–Raskin, Proof of the geometric Langlands conjecture I, §1.1, identifies this line with the canonical line up to a constant one-dimensional factor, and defines its automorphic category by the $\mu_2$-gerbe of square roots of (7.1): \[ D\text{-}\mathrm{mod}_{1/2}(\mathrm{Bun}_G). \tag{7.2} \] These determinant and moduli-stack assertions are used as cited inputs.

Locally this agrees with the half-TDO description. On an étale cover choose a square root $P$ with $P^2=\det_{\mathrm{Bun}_G}$. Differential operators on $P$ change by $d\log$ of its transitions, which is one half the logarithmic differential of the determinant transitions. The local equivalence sends a $\operatorname{Diff}(P,P)$-module $M$ to the ordinary D-module $P^{-1}\otimes M$; its inverse is tensoring with $P$, by (3.2). On the root gerbe the automorphism $-1$ of $P$ acts by $-1$ on $P^{-1}\otimes M$. Changing a chosen root changes these identifications by the flat two-torsion line of the two roots. The overlap maps and their triple-overlap signs therefore give exactly descent with this $\mu_2$-character. This proves agreement with the half-twisted category. The gerbe formulation makes the global twisting datum explicit, while (3.1) calculates its local infinitesimal operators.

A choice of a square root of $\omega_C$ supplies a square root of the determinant line, as cited there from Beilinson–Drinfeld, §4. That choice gives an equivalence between (7.2) and an untwisted category, but the equivalence depends on the choice. The closing remark of §1.1 retains the twisted formulation for compatibility with the Hecke action. In particular “half-twisted” is not an instruction to tensor globally with an unspecified line bundle.

8. Exercises and solutions

Exercise 15.1 (easy). Prove that every strongly $\mathbb G_m$-equivariant D-module on $\mathbb G_m$, with translation action, is a direct sum of copies of $\mathcal O_{\mathbb G_m}$.

Solution. The action makes $\mathbb G_m\to\mathrm{pt}$ a principal bundle. Theorem 2.1 descends the module to a vector space $V$ and identifies its pullback with $\mathcal O_{\mathbb G_m}\otimes_k V$, with differentiation on the first factor. Choosing a basis of $V$ writes it as a direct sum. The tensor expression is canonical; the basis decomposition need not be. For coherent modules $V$ is finite dimensional. This also proves that an integral Laurent Euler connection admits this form.

Exercise 15.2 (easy). Give a finite-group example in which equivariance is extra structure.

Solution. Let the group of order two act on a point. Its Lie algebra is zero, so both its trivial and sign representations on $k$ are strong. Their underlying D-module is the same one-dimensional vector space. The identity map between those vector spaces is not equivariant, since it would have to intertwine $1$ and $-1$. Thus the forgetful functor is not full, and an underlying module does not determine its linearization. More generally the point quotient has abelian category $\operatorname{Rep}(G)$ for finite $G$.

Exercise 15.3 (medium). Compute $\Gamma(\mathbb P^1,\mathcal D_{\mathcal O(n)})$ for every integer $n$.

Solution. The notation means differential operators on the line bundle $\mathcal O(n)$, hence $\mathcal D_{\mathcal O(1)}^n$ by (3.2). Equations (3.5)–(3.7), including the associated-graded surjectivity and injectivity proof, give \[ U(\mathfrak{sl}_2)/(h^2+2h+4fe-n(n+2)). \] This answer holds for negative $n$ as well. Its filtered piece of exact symbol degree $m$ has dimension $2m+1$; therefore the ring is infinite dimensional. If $n\geq0$, its action on the $(n+1)$-dimensional space of sections of $\mathcal O(n)$ is the explicit highest-weight action calculated after (3.7). That representation does not replace the ring in the answer.

Exercise 15.4 (medium). Describe the strongly scaling-equivariant D-modules on $\mathbb A^1$ by linear algebra, including morphisms and the coherence condition.

Solution. The answer is two spaces $V_0,V_1$, arrows $a:V_1\to V_0$, $b:V_0\to V_1$, and both relations $ab=ba=0$. The forward construction is (5.1)–(5.2), and the inverse is (5.3). All nonexceptional weights are identified by invertible $x$ or $\partial$ maps. A D-map preserves $\theta$-weights, so it is determined by maps $V_i\to V_i'$ commuting with the two arrows; conversely such maps extend using (5.3). Finite-dimensional spaces give and are necessary for coherent modules, by the proof in Section 5. Dropping either relation gives an incorrect answer: take $V_0=k$, $V_1=k^2$, $a=(1,0)$, $b=(0,1)^t$. Then $ab=0$ but $ba$ is a nonzero square-zero operator, which violates semisimplicity at weight $-1$.

Exercise 15.5 (hard). For connected affine $G$, identify the abelian D-module category on $BG$ and compute the self-Ext algebra of the constant object in the derived quotient category. Explain the case $G=\mathbb G_m$ using Arinkin–Gaitsgory.

Solution. Strong representations on a point have zero derivative and hence trivial connected-group action, proving (6.1) with all vector spaces allowed. The derived smooth-descent definition gives the de Rham totalization (6.2), so self-Ext is $H^*_{\rm dR}(BG)$, or $H^*(BG^{\rm an},\mathbb C)$ over $\mathbb C$. The identification respects products because the Spencer self-Hom complex and the descent maps respect composition. For $\mathbb G_m$ the nerve calculation gives a polynomial generator of degree two, as in (6.3). The derived-intersection equivalence (6.4) places the constant object at the augmentation module of the exterior DG algebra; the explicit resolution (6.5) gives the same polynomial Yoneda algebra and shows why the constant object is not compact. The abelian category of vector spaces has no positive Ext, so its ordinary derived category is not the answer to this exercise.

What this lesson does not prove

We use effective smooth descent and basic line-bundle cohomology as algebraic geometry prerequisites. The derived descent formalism is defined and used here; its general construction and functoriality belong to D-modules on stacks, ind-schemes and the de Rham prestack. The algebraic de Rham comparison used in (6.2) is the theorem stated with its primary locator in The de Rham functor. The general classification of compact modules over a DG algebra as retracts of finite semifree modules is a derived-category prerequisite.

We do not prove the derived-intersection equivalence (6.4), the loop-group uniformization or its central extension, the canonical determinant-line identification on $\mathrm{Bun}_G$, or the existence of its root from a square root on the curve. Their locators are, respectively, Arinkin–Gaitsgory, §11.2; Frenkel, §§7.4–7.5; and Gaitsgory–Raskin, §1.1, with its reference to Beilinson–Drinfeld, §4. The explicit root-gerbe/local-TDO agreement uses the descent calculation given in Section 7. No localization equivalence, Hecke theorem or geometric Langlands theorem is inferred from the adjunction alone.

References

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