Generic Stationary Local Character

• Generic stationary local character is a framework where a local object, combined with conditions of genericity and stationarity, yields canonical invariants across multiple mathematical domains.
• It appears in geometric representation theory via multiplicative local systems, in model theory through club subsets for non-forking, and in probability with translation-invariant local functionals.
• The approach underpins key equivalences such as t-exact parabolic induction and higher Deligne–Lusztig correspondences, highlighting its structural and practical significance.

Searching arXiv for papers relevant to "generic stationary local character" and closely related uses of the phrase. “Generic stationary local character” does not appear in the listed literature as a single universally standardized formal definition. Instead, it names a recurrent structural pattern in which locality, genericity, and a stationary or large-set condition organize a character-like object, an independence relation, or a local invariant. In geometric representation theory, the closest formal realization is the geometric avatar of a generic stationary local character on a torus or Levi subgroup, where a sufficiently nontrivial multiplicative local system determines an idempotent summand and a generic subcategory on which positive-depth parabolic induction becomes a $t$-exact equivalence (Bezrukavnikov et al., 2024). In model theory, “local character” is phrased through club and stationary subsets of $[M]^{|T|}$ and characterizes NSOP$_1$ over models (Kaplan et al., 2017). In AEC theory, weak local character is strengthened to full local character inside a good frame (Jarden et al., 2011). In probability, local intrinsic location functionals and stationary local random countable sets provide separate stationary-local frameworks (Shen, 2014, Vidmar et al., 2022).

1. Terminological status and scope

The term is best understood as a family resemblance rather than a single doctrine. Some papers explicitly use it only heuristically. The mod-$\ell$ paper on local character expansion says that its construction is “closely related to the idea of a ‘generic stationary local character’ only in a loose structural sense” and also states that it “does not formulate or prove any genericity/stationarity theorem in the usual sense” (Tsai, 23 Oct 2025). Likewise, the paper on the explicit LLC for $G_2$ states that it does not introduce a special term “generic stationary local character” as a formal definition, but instead studies local character germs on neighborhoods of semisimple or topologically unipotent elements (Suzuki et al., 2023).

This usage pattern separates at least three technical meanings. First, there is a geometric-representation-theoretic meaning, where genericity is imposed on local systems or coadjoint data and locality is encoded by sheaf-theoretic support or induction from parahoric or finite-level subgroups. Second, there is a model-theoretic meaning, where “local character” is a structural property of independence, and “stationary” refers to stationary or club subsets of a natural poset of small submodels. Third, there is a probabilistic meaning, where stationarity is literal translation-invariance and locality is encoded by interval-wise dependence or measurability.

A common misconception is that every paper involving “local character” or “stationary” is addressing the same notion. The listed works show otherwise. Some use the phrase as a geometric or structural analogy; others use “stationary” in the set-theoretic sense of stationary subsets; others use it in the ergodic or stochastic-process sense.

2. Geometric realization on parahoric group schemes

The most direct formal realization occurs in the theory of positive-depth character sheaves on parahoric group schemes. For a connected reductive group over $\breve F$, a point $x$ in the Bruhat–Tits building, and $r>0$, the Moy–Prasad filtration gives a smooth affine group scheme $G_{x,r}$, and the paper writes

$G_r := G_{0:r+}.$

If $[M]^{|T|}$0 is a parabolic with $[M]^{|T|}$1, then one has

$[M]^{|T|}$2

A multiplicative local system on an algebraic group $[M]^{|T|}$3 is a rank-1 local system $[M]^{|T|}$4 such that

$[M]^{|T|}$5

For a generic element $[M]^{|T|}$6, the paper defines

$[M]^{|T|}$7

extends them to $[M]^{|T|}$8 and $[M]^{|T|}$9, and proves that they are idempotent for convolution: $_1$0 The associated generic subcategories are

$_1$1

The genericity condition requires that $_1$2 not vanish on any root space outside $_1$3, and that its stabilizer in the Weyl group be exactly that of $_1$4. The paper identifies this as the geometric avatar of a generic stationary local character: the local system is sufficiently nontrivial along all directions transverse to the Levi (Bezrukavnikov et al., 2024).

The central theorem is that positive-depth parabolic induction becomes an equivalence on this generic locus. With

$_1$5

the paper proves that

$_1$6

is a $_1$7-exact equivalence, with inverse

$_1$8

On the generic locus, $_1$9, the corresponding restriction functors agree after convolution with $\ell$0, perversity is preserved, and simple perverse sheaves are sent to simple perverse sheaves. This is the categorical content of “stationary local character” in this setting: generic data carve out a stable categorical chamber in which induction is invertible and exact (Bezrukavnikov et al., 2024).

The iterative construction of a generic datum

$\ell$1

produces a simple $\ell$2-equivariant perverse sheaf $\ell$3, yielding a family of positive-depth character sheaves. In the toral case $\ell$4, $\ell$5, a sufficiently generic multiplicative local system on $\ell$6 gives

$\ell$7

which resolves Lusztig’s conjecture in positive depth. Under a mild condition on $\ell$8, the Frobenius trace matches parahoric Deligne–Lusztig induction: $\ell$9 Accordingly, the “character” aspect is not merely parameterizing data: it is realized by simple perverse sheaves whose traces recover representation-theoretic characters (Bezrukavnikov et al., 2024).

3. Generic character sheaves over finite local rings

A related realization appears for reductive groups over finite local rings at even level. In that setting one works with a smooth affine group scheme $G_2$0 over $G_2$1, with $G_2$2 a fixed positive even integer, and after the Greenberg functor obtains an algebraic group

$G_2$3

with Frobenius $G_2$4 and

$G_2$5

For a maximal torus $G_2$6 and a character $G_2$7, the paper distinguishes general position, regularity, and genericity, where generic means regular plus general position plus a stabilizer condition on the trivial extension $G_2$8 to $G_2$9. The higher Deligne–Lusztig representation is

$\breve F$0

and for generic $\breve F$1 one has

$\breve F$2

The paper explicitly presents this as its version of a “Generic Stationary Local Character”: a character built geometrically from a perverse complex whose Frobenius trace is the higher Deligne–Lusztig character (Chen, 2016).

The geometric construction uses

$\breve F$3

with maps

$\breve F$4

For the rank-one local system $\breve F$5 on $\breve F$6, the generic character sheaf is

$\breve F$7

Its characteristic function is

$\breve F$8

and the key identity is

$\breve F$9

This packages locality in the correspondence $x$0, genericity in the conditions on $x$1, and character in the sheaf-function correspondence (Chen, 2016).

The paper conjectures that if $x$2 is geometrically generic, then

$x$3

is a simple perverse sheaf, and proves this for $x$4. It also develops induction, restriction, and Frobenius reciprocity for these complexes. In this finite-ring setting, the phrase therefore denotes a fully geometric local-character package: local systems on $x$5, induced sheaf complexes on $x$6, and Frobenius traces equal to higher Deligne–Lusztig characters (Chen, 2016).

4. Stationary local character in model theory and AECs

In model theory, “stationary local character” has a genuinely set-theoretic meaning. For NSOP$x$7 theories, the local character of Kim-independence is expressed in terms of club and stationary subsets of $x$8. If $x$9 and $r>0$0 is a type over $r>0$1, the paper proves that the collection of elementary substructures of size $r>0$2 over which $r>0$3 does not Kim-fork is a club of $r>0$4, and that this property characterizes NSOP$r>0$5 (Kaplan et al., 2017). More precisely, the equivalent formulations include the statement that

$r>0$6

is a club in $r>0$7. The paper also proves a dual local-character theorem for dual types, again with stationary and club formulations.

This reframes local character away from base monotonicity. In simple theories, forking local character is often expressed by shrinking to a small base and using monotonicity. For Kim-independence in NSOP$r>0$8, base monotonicity fails in general unless the theory is simple, so the replacement is not a cone of smaller bases but a club of good bases. The paper describes this as a kind of stationary or generic local character over models (Kaplan et al., 2017).

An older AEC result treats a different but related problem. Starting from Shelah’s non-forking relation on an almost good $r>0$9-frame, the paper “Weakening the local character” upgrades weak local character to full local character. The local character axiom states that for every limit ordinal $G_{x,r}$0, if $G_{x,r}$1 is an increasing continuous sequence of models in $G_{x,r}$2 and

$G_{x,r}$3

then there exists $G_{x,r}$4 such that the type does not fork over $G_{x,r}$5. The earlier weak local character only guaranteed such behavior along $G_{x,r}$6-increasing chains. The paper proves that the induced relation $G_{x,r}$7 yields a full good $G_{x,r}$8-frame, so the resulting non-forking calculus satisfies the main properties of non-forking in superstable first-order theories (Jarden et al., 2011).

Across these two model-theoretic settings, “stationary local character” refers not to characters of representations but to the largeness of the set of local bases witnessing independence. The stationary component is literal stationarity in the club/stationary-set sense, and the local component is the existence of a small elementary submodel or earlier stage in a chain over which the type stabilizes.

5. Probabilistic stationary-local frameworks

In stochastic-process theory, a distinct stationary-local theory is developed through intrinsic location functionals. For a path space $G_{x,r}$9 closed under translations and compact non-degenerate intervals $G_r := G_{0:r+}.$0, an intrinsic location functional is a map

$G_r := G_{0:r+}.$1

satisfying measurability, interval-valuedness, shift compatibility, stability under restrictions, and consistency of existence. The paper then introduces a local version $G_r := G_{0:r+}.$2 for fixed $G_r := G_{0:r+}.$3, defined on windows $G_r := G_{0:r+}.$4 with a local consistency axiom replacing the global restriction axioms (Shen, 2014).

The main stationary characterization is that for a continuous stochastic process $G_r := G_{0:r+}.$5, stationarity is equivalent to the condition that for any $G_r := G_{0:r+}.$6 and any local intrinsic location functional $G_r := G_{0:r+}.$7, the law of

$G_r := G_{0:r+}.$8

does not depend on $G_r := G_{0:r+}.$9. Equivalently, $[M]^{|T|}$00 has a density on $[M]^{|T|}$01 satisfying the total variation constraints. The paper also proves that every such location can be represented as the maximal element of a random partially ordered set over the interval, and that as the interval slides the chosen location changes in a piecewise linear way described by Theorem 5.1 (Shen, 2014).

A related Wiener-noise theory studies stationary local random countable sets. A random countable set $[M]^{|T|}$02 on two-sided Wiener space is local if for every $[M]^{|T|}$03, $[M]^{|T|}$04 admits an $[M]^{|T|}$05-measurable enumeration, and stationary if

$[M]^{|T|}$06

The paper proves that the Brownian local minima, maxima, and extrema are stationary local dense random countable sets, and constructs continuum many additional examples $[M]^{|T|}$07, $[M]^{|T|}$08, arising from zero sets of squared Bessel diffusions. For distinct $[M]^{|T|}$09,

$[M]^{|T|}$10

It further develops honest indexation, proves a splitting theorem analogous to Wiener–Hopf factorization, shows that honestly indexed sets avoid stopping times, and establishes minimality for honestly indexed dense stationary local sets (Vidmar et al., 2022).

These probabilistic theories do not use “character” in the representation-theoretic sense. The common structure lies instead in a local object determined from a moving window or local sigma-field and constrained by translation-invariant stationarity.

6. Related notions, non-examples, and boundaries of the term

Several nearby literatures are explicitly adjacent but not identical. The mod-$[M]^{|T|}$11 local character expansion paper defines coefficients $[M]^{|T|}$12 by expressing the multiplicity vector of degenerate Moy–Prasad types,

$[M]^{|T|}$13

in the basis of nilpotent-orbit vectors $[M]^{|T|}$14, leading to

$[M]^{|T|}$15

The paper stresses, however, that this is a stable expansion framework rather than a theorem about genericity or stationarity in the usual sense (Tsai, 23 Oct 2025).

The paper on the explicit LLC for $[M]^{|T|}$16 studies Harish-Chandra local character theory, Adler–Korman expansions near tame semisimple elements, and stability of $[M]^{|T|}$17-packet sums. Its “local character” is the locally constant character function on neighborhoods of semisimple or topologically unipotent elements, and the stable combinations are detected by cancellation of unstable Green-function terms. This is local character theory in the representation-theoretic sense, but not a formal theory of generic stationary local character (Suzuki et al., 2023).

Other uses of “generic,” “stationary,” and “local” are further removed. The paper on non-generic character stacks studies prescribed semisimple local monodromies and their effect on the cohomology of character stacks; genericity there concerns the absence of proper compatible invariant subspaces, not a stationary local character principle (Scognamiglio, 2023). The active-matter paper on “Generic nonlocal statistics of the stationary measure in conserved active systems” addresses stationary measures, nonanalytic Fourier kernels, and the failure of a local Landau–Ginzburg expansion; despite the co-occurrence of “generic,” “stationary,” and “local,” its subject is a perturbative theory of nonequilibrium stationary measures rather than local character in any representation-theoretic or model-theoretic sense (Luca et al., 12 Jun 2026).

Taken together, these boundaries suggest that the phrase is most precise when the surrounding paper explicitly ties genericity to a local object and a stable or stationary regime. Where that tie is absent, the phrase is best treated as analogical rather than terminological.

7. Conceptual synthesis

Across the listed works, three recurrent ingredients appear. The first is a local datum: a multiplicative local system on a torus or Levi, a type over a model, a local intrinsic location functional, or a local random countable set. The second is a largeness or stability condition: genericity of $[M]^{|T|}$18 or $[M]^{|T|}$19, a club subset of $[M]^{|T|}$20, or translation-invariant stationarity under shifts. The third is an output that is rigid enough to be canonical: a generic subcategory and simple perverse sheaf, a non-Kim-dividing base, a full good frame, a distributional characterization of stationarity, or a minimal stationary local set.

In geometric representation theory, this synthesis is strongest. A generic stationary local character is what isolates the summand

$[M]^{|T|}$21

preserves perversity, and aligns character sheaves with parahoric Deligne–Lusztig theory (Bezrukavnikov et al., 2024). In the finite-local-ring setting, it is the local system $[M]^{|T|}$22 whose induced complex $[M]^{|T|}$23 has characteristic function equal to the higher Deligne–Lusztig character (Chen, 2016). In model theory, it is the club-sized family of small bases over which Kim-dividing disappears (Kaplan et al., 2017). In stochastic settings, it is the local rule whose translation-invariant law detects stationarity (Shen, 2014).

The phrase therefore denotes not a single invariant but a shared architecture: a local object, a generic or stationary regime, and a resulting exactness, rigidity, or canonical expansion. This suggests that the enduring content of “generic stationary local character” is structural rather than lexical.