Geometric Eisenstein Functors

• Geometric Eisenstein series functors are sheaf-theoretic analogues that use parabolic pull–push operations to transfer sheaves between moduli stacks of bundles.
• They incorporate kernels, compactifications, and normalization twists to address nonproper maps and maintain duality and exactness in complex geometric settings.
• These functors bridge the automorphic and spectral sides via Hecke structures and adjunctions, playing a key role in the geometric Langlands program and Fargues–Fontaine theory.

Geometric Eisenstein series functors are sheaf-theoretic analogues of classical Eisenstein series and parabolic induction. For a reductive group $G$, a parabolic subgroup $P\subset G$ with Levi quotient $M$, and the correspondence of moduli stacks of bundles

$\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$

they are defined by pull–push operations between sheaf categories on $\Bun_M$ and $\Bun_G$. Depending on the context, one obtains $!$- and $*$-versions, compactified variants, and normalized forms adapted to D-modules, $\ell$-adic sheaves, genuine metaplectic categories, and sheaves on stacks of bundles over the Fargues–Fontaine curve. Their structure is controlled by kernels on $\Bun_P$ or its compactifications, and their significance is tied to constant term functors, Hecke actions, Verdier duality, Whittaker functionals, and spectral Eisenstein functors in geometric Langlands (Drinfeld et al., 2013, Taylor, 2022, Hamann et al., 2024, Lysenko, 2012).

1. Basic correspondence and principal definitions

The foundational input is always the moduli-theoretic correspondence attached to a parabolic reduction. On a smooth projective curve over an algebraically closed field, Drinfeld–Gaitsgory formulate four basic D-module functors

$P\subset G$0

with the expected adjunctions $P\subset G$1 and $P\subset G$2 once the relevant domains of definition are established (Drinfeld et al., 2013). Taylor uses the same pull–push pattern on D-modules over a complex curve and writes the automorphic Eisenstein functor as

$P\subset G$3

with the remark that one may normalize by twisting along $P\subset G$4 and by cohomological shifts without changing the construction up to an explicit line bundle (Taylor, 2022).

In metaplectic rank two, Lysenko replaces the untwisted kernel by a genuine perverse sheaf. For $P\subset G$5, with $P\subset G$6 and $P\subset G$7, the functor is the integral transform

$P\subset G$8

where $P\subset G$9 is the perverse extension of a rank-one Artin–Schreier local system twisted by the genuine $M$0-character; here “genuine” means odd for the central $M$1-action (Lysenko, 2012).

On the Fargues–Fontaine curve, Hamann–Hansen–Scholze define normalized $M$2-adic Eisenstein functors

$M$3

together with the corresponding constant term functors defined by the same kernel $M$4 (Hamann et al., 2024). In the Borel case over the Fargues–Fontaine curve, a further normalization gives the principal-series functor

$M$5

where $M$6 is the character sheaf for $M$7 (Hamann, 2022).

Setting	Eisenstein functor	Kernel or normalization
D-modules on $M$8	$M$9, $\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$0	Pull–push along $\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$1 (Drinfeld et al., 2013)
Genuine $\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$2-adic sheaves for $\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$3	$\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$4	Genuine kernel $\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$5 (Lysenko, 2012)
Fargues–Fontaine, general $\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$6	$\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$7	Self-dual $\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$8 (Hamann et al., 2024)
Fargues–Fontaine, Borel	$\Bun_M \xleftarrow{\,q\,} \Bun_P \xrightarrow{\,p\,} \Bun_G,$9	$\Bun_M$0 (Hamann, 2022)

A recurring source of confusion is that “the” geometric Eisenstein functor is not a single canonical normalization. The literature distinguishes $\Bun_M$1- and $\Bun_M$2-versions, compactified and non-compactified kernels, and various twists by modulus or Whittaker normalizations. The common core is the parabolic pull–push geometry.

2. Kernels, compactifications, and normalization data

The kernel perspective is central. In Drinfeld–Gaitsgory, the noncompact map $\Bun_M$3 is not proper in general, so the existence of $\Bun_M$4 is nontrivial. The key point is that $\Bun_M$5 is nevertheless well-defined on the essential image of $\Bun_M$6, which yields a continuous functor $\Bun_M$7 on each connected component of $\Bun_M$8 (Drinfeld et al., 2013).

Compactification enters through Drinfeld’s compactified moduli of parabolic reductions. Færgeman–Hayash consider a relative compactification

$\Bun_M$9

whose fibers are stratified by defect $\Bun_G$0, and define the compactified Eisenstein functor

$\Bun_G$1

This makes the compactification itself part of the kernel formalism rather than merely an auxiliary technical device (Faergeman et al., 18 Jul 2025).

In the metaplectic $\Bun_G$2 setting, the kernel is already nonstandard on the open locus $\Bun_G$3 where the $\Bun_G$4-reduction is a subbundle. Lysenko defines $\Bun_G$5 as the perverse extension of the rank-one Artin–Schreier local system $\Bun_G$6 arising from the $\Bun_G$7-torsor structure $\Bun_G$8, twisted by the genuine $\Bun_G$9-character. The theorem that $!$0 is ULA with respect to $!$1 is then the structural input behind duality and exactness properties of $!$2 (Lysenko, 2012).

The Fargues–Fontaine formalism packages the normalization directly into the dualizing object. Hamann–Hansen–Scholze define

$!$3

so that $!$4 is Verdier-self-dual, while Caraiani–Hansen–Jin–Le Bras–Scholze identify in the Borel case the dualizing object on $!$5 with $!$6, producing the self-dual $!$7 used in $!$8 (Hamann et al., 2024, Hamann, 2022).

These constructions show that the kernel is not merely a bookkeeping device. It encodes the normalizing twist, the compactification data, and the precise duality behavior of the functor.

3. Adjunctions, exactness, and functional equations

The first general structural result is the restoration of the four-functor pattern. Drinfeld–Gaitsgory prove that $!$9 exists and is canonically isomorphic to the opposite-parabolic functor $*$0. Equivalently, $*$1 is an adjoint pair, and Verdier duality intertwines $*$2 with $*$3 (Drinfeld et al., 2013).

Gaitsgory’s later analysis shows that on the non-quasi-compact stack $*$4, ordinary Verdier duality is replaced by the pseudo-identity or “miraculous duality” functor

$*$5

which is an equivalence. Relative to this duality, Eisenstein and constant term satisfy a “strange” functional equation, for example

$*$6

where $*$7 (Gaitsgory, 2014). This identifies the functional equation for Eisenstein series with a categorical statement about duality on $*$8.

In the metaplectic case, Lysenko proves a parallel package of properties. Since $*$9 is ULA over $\ell$0, the functor $\ell$1 commutes with Verdier duality, preserves purity of weights, and is perverse $\ell$2-exact up to the shift $\ell$3. The same section also establishes factorization-type compatibility and transitivity with respect to parabolic induction, such as

$\ell$4

for appropriate nested Levi data (Lysenko, 2012).

Over the Fargues–Fontaine curve, Hamann–Hansen–Scholze prove finiteness results analogous to classical properties of parabolic induction and Jacquet modules. In particular, $\ell$5 preserves compact objects, $\ell$6 preserves ULA objects, there is a canonical isomorphism $\ell$7, and geometric Bernstein second adjointness gives

$\ell$8

Equivalently, $\ell$9 and $\Bun_P$0 (Hamann et al., 2024).

Taken together, these results show that geometric Eisenstein functors are not isolated pull–push constructions. They sit inside a rigid system of adjunctions and dualities, and much of the subtlety comes from making these structures survive nonproperness and non-quasi-compactness.

4. Whittaker models and Fourier coefficients

Whittaker theory provides one of the sharpest probes of geometric Eisenstein series. In the metaplectic $\Bun_P$1 case, Lysenko introduces the stack $\Bun_P$2 of extensions

$\Bun_P$3

the evaluation map $\Bun_P$4, the Artin–Schreier sheaf $\Bun_P$5, and the map $\Bun_P$6. The most non-degenerate Whittaker coefficient of a genuine automorphic sheaf $\Bun_P$7 is then

$\Bun_P$8

For a rank-one local system $\Bun_P$9 with automorphic sheaf $P\subset G$00, the Whittaker coefficient of $P\subset G$01 is computed on each $P\subset G$02 by a direct sum involving

$P\subset G$03

recovering the classical Maas–Waldspurger formula in geometric guise. In particular, the nonzero coefficients occur only in the correct even degrees (Lysenko, 2012).

The same paper develops a family version of the Whittaker category for $P\subset G$04. Auxiliary stacks $P\subset G$05, $P\subset G$06, and $P\subset G$07 encode generic sections and generic Hecke $P\subset G$08-equivariance along the unipotent radical. A key theorem constructs an exact equivalence

$P\subset G$09

and identifies standard perverse sheaves $P\subset G$10 with the Whittaker kernel $P\subset G$11 on the relevant strata. The paper emphasizes that these calculations lead to a conjectural characterization of the Whittaker sheaf for $P\subset G$12, though its existence is not clear (Lysenko, 2012).

Taylor proves a general Whittaker-coefficient formula for geometric Eisenstein series in the D-module setting. For a $P\subset G$13-local system $P\subset G$14 on $P\subset G$15 and a Hecke eigensheaf $P\subset G$16 with eigenvalue $P\subset G$17,

$P\subset G$18

The proof passes through Zastava spaces, factorization homology on the Ran space, the chiral enveloping algebra $P\subset G$19, and the Beilinson–Drinfeld formula

$P\subset G$20

which yields Lie-algebra homology as functions on the moduli of $P\subset G$21-local systems (Taylor, 2022).

A plausible implication is that Whittaker coefficients organize the discrepancy between the automorphic pull–push definition and the spectral geometry of $P\subset G$22. In the cited works this is not merely heuristic: it becomes an explicit identification.

5. Hecke structures, parabolic functoriality, and spectral compatibilities

Hecke-equivariance is a decisive refinement of the basic functoriality. In the metaplectic $P\subset G$23 construction, the ULA property of $P\subset G$24 is obtained by building twisted Hecke correspondences over $P\subset G$25 and proving Braverman–Gaitsgory equivariance for all Hecke moves at finitely many points. The same framework yields factorization and transitivity for Eisenstein functors as parabolic data vary (Lysenko, 2012).

Færgeman–Hayash generalize the principal-parabolic Hecke compatibility of Braverman–Gaitsgory to arbitrary parabolics. Their key result is that the compactified kernel

$P\subset G$26

admits a canonical lift to a $P\subset G$27-Hecke module, expressed by compatible isomorphisms

$P\subset G$28

On the open locus $P\subset G$29, the kernel $P\subset G$30 carries an enhanced Drinfeld–Plücker structure

$P\subset G$31

compatible with the algebra structure on the Chevalley complex $P\subset G$32. As a consequence, both $P\subset G$33 and $P\subset G$34 are $P\subset G$35-linear (Faergeman et al., 18 Jul 2025).

The same work relates compactified and non-compactified kernels through semi-infinite geometry and Koszul duality. The semi-infinite IC-sheaf $P\subset G$36 carries a coaction of $P\subset G$37, and taking invariants recovers $P\subset G$38. Globalizing this comparison yields a Koszul-duality relation between the kernels

$P\subset G$39

hence between the corresponding functors: $P\subset G$40 This identifies the open and compactified Eisenstein constructions as dual manifestations of the same parabolic data (Faergeman et al., 18 Jul 2025).

On the spectral side, Taylor defines

$P\subset G$41

while Færgeman–Hayash formulate compatibilities with the automorphic-to-spectral geometric Langlands functor

$P\subset G$42

namely

$P\subset G$43

In their presentation, this compatibility is deduced by passing through Whittaker models and the geometric Casselman–Shalika equivalence (Taylor, 2022, Faergeman et al., 18 Jul 2025).

6. Fargues–Fontaine theory, local Langlands applications, and categorical decomposition

The Fargues–Fontaine curve supplies a local analogue of the global theory. Caraiani–Hansen–Jin–Le Bras–Scholze construct the normalized Borel Eisenstein functor $P\subset G$44 on $P\subset G$45 for a quasi-split connected reductive group $P\subset G$46 with simply connected derived subgroup. They prove a filtered Hecke-eigenproperty: for $P\subset G$47, the complex $P\subset G$48 carries a $P\subset G$49-equivariant filtration indexed by weight tuples $P\subset G$50, and its graded pieces are

$P\subset G$51

up to the stated shift and Tate twist. For a toral parameter $P\subset G$52, this implies that $P\subset G$53 behaves as a Hecke eigensheaf with eigenvalue determined by the weight decomposition of $P\subset G$54 (Hamann, 2022).

They also compute stalks on Harder–Narasimhan strata. Under mild genericity and compatibility assumptions, for unramified $P\subset G$55,

$P\subset G$56

and the sheaf vanishes on non-unramified strata. From the top-stratum description one obtains canonical intertwining isomorphisms

$P\subset G$57

The same package recovers special cases of Shin’s averaging formula and refines it to non-minuscule situations, including explicit cohomological degrees and Tate twists (Hamann, 2022).

Hamann–Hansen–Scholze develop a broader Fargues–Fontaine theory of geometric Eisenstein and constant term functors. They prove preservation of compactness and ULA, geometric second adjointness, and a cuspidal–Eisenstein decomposition of $P\subset G$58. The Eisenstein subcategory is generated under colimits by the essential images of $P\subset G$59 for proper parabolics, while the cuspidal subcategory is the joint kernel of all $P\subset G$60. Their theorem gives a semiorthogonal decomposition

$P\subset G$61

and for $P\subset G$62 an orthogonal decomposition

$P\subset G$63

In the example $P\subset G$64, they identify $P\subset G$65 with compactly supported parabolic induction on the trivial stratum and $P\subset G$66 with the normalized Jacquet module on generic strata (Hamann et al., 2024).

This local theory shows that geometric Eisenstein functors are not confined to the global setting of algebraic curves over algebraically closed fields. In the Fargues–Fontaine context they become tools for local Langlands, for the geometry of Harder–Narasimhan strata, and for categorical decompositions that parallel the Bernstein theory of smooth $P\subset G$67-adic representations.