GL-equivariant Bilinear Incidence Varieties

• GL-equivariant bilinear incidence varieties are defined by imposing bilinear constraints invariant under the GL-action, unifying the study of associative, commutative, Leibniz, and Lie algebras.
• The construction utilizes ambient spaces of multilinear maps and flag varieties, providing explicit quadratic equations that link geometric orbits with cohomological properties.
• Applications include canonical bilinear pairings in the Rankin–Selberg method and the analytic continuation of L-functions, bridging algebraic geometry with representation theory.

GL-equivariant bilinear incidence varieties provide a unified geometric framework to paper algebraic and representation-theoretic structures invariant under the general linear group. This theory interrelates the geometry of orbits in spaces of algebra laws, the construction of canonical bilinear pairings (such as those arising in the Rankin–Selberg method), and the moduli and deformation theory of associative, commutative, Leibniz, and Lie algebras. Central to this framework are incidence varieties defined by bilinear constraints, with a natural action of $\mathrm{GL}(n)$ or $\mathrm{GL}(V)$ ensuring the invariance and equivariance properties essential for applications in both geometric representation theory and algebraic geometry.

1. Ambient Spaces and GL-action

The foundational objects are affine spaces of multilinear maps: $$V^{2,1} = \mathrm{Hom}_k(V \otimes V, V) \cong k^{n^3}$$, parameterizing all bilinear multiplications on a finite-dimensional $k$-vector space $V \cong k^n$, and $$V^{3,1} = \mathrm{Hom}_k(V^{\otimes 3}, V) \cong k^{n^4}$$ for trilinear maps. The group $\mathrm{GL}(V)$ acts on $V^{2,1}$ by transport of structure: $$(g \cdot x)_{i,j}^\ell := \text{coeff}_{e_\ell} \left( g(\mu_x(g^{-1}e_i, g^{-1}e_j)) \right)$$ for $g \in \mathrm{GL}(V)$ and structure constants $x_{i,j}^\ell$ for $\mu_x$.

In representation-theoretic applications, such as the Rankin–Selberg method, the ambient space is a product of flag varieties—e.g., $$(B_n \backslash \mathrm{GL}_n)^2 \times k^{1 \times n}$$ for $\mathrm{GL}_n \times \mathrm{GL}_n$, with right action on each factor and vector by simultaneous conjugation and linear automorphism, ensuring GL-equivariance of the associated orbits and incidence relations (Li et al., 2021, Kaygun, 21 Nov 2025).

2. Definition and Construction of Bilinear Incidence Varieties

A bilinear incidence variety is constructed by imposing bilinear relations reflecting algebraic laws or transversality conditions. The paradigm example is the "bilinearization" of the associator, defining

$\beta: V^{2,1} \times V^{2,1} \to V^{3,1}$

by $$\beta(x, y)(a, b, c) = \mu_x(\mu_y(a, b), c) - \mu_x(a, \mu_y(b, c))$$. In coordinates,

$$\beta^m_{i,j,k}(x, y) = \sum_\ell (x_{i,j}^\ell y_{\ell,k}^m - y_{i,\ell}^m x_{j,k}^\ell)$$

The variety is then

$$I = \{ (x, y) \in V^{2,1} \times V^{2,1} \mid x^T q y = 0 \ \forall\, q \in Q^{\mathrm{sym}} \}$$

where $Q^{\mathrm{sym}}$ is the span of symmetric components obtained by dualization. The construction is equivariant under $\mathrm{GL}(V)$ action, rendering $I$ a $\mathrm{GL}(V)$-stable closed subscheme (Kaygun, 21 Nov 2025).

In representation theory, the incidence variety is realized as the unique open orbit $\Omega$ in a product of flag varieties—parametrizing, for instance, configurations $(F_1, F_2, v)$ of two flags and a vector in $k^n$ such that $F_1^{n-1} \oplus \langle v \rangle = V$ and $v \not\in F_1^{n-1} \cup F_2^{n-1}$ (Li et al., 2021).

3. Diagonal Slices and Algebra Law Varieties

Diagonal embeddings of the incidence variety recover classical algebra law varieties. On the diagonal $(x, x)$, the relation $\beta(x, x) = 0$ encodes associativity; more generally:

• For associative algebras, $Alg(V) = \{ x \in V^{2,1} \mid \Phi(x) = 0 \}$, where $\Phi(x) = \beta(x, x)$.
• For commutative associative algebras, add symmetry constraints $x_{i, j}^\ell = x_{j, i}^\ell$.
• For Leibniz algebras, use $B(x, x) = 0$ for an appropriate bilinearization $B$.
• For Lie algebras, further require skew-symmetry $x_{i, j}^\ell = -x_{j, i}^\ell$ so that $B(x, x)$ encodes the Jacobi identity.

These varieties are cut out by explicit quadratic equations in the structure constants and are stable under $\mathrm{GL}(V)$ (Kaygun, 21 Nov 2025).

4. Fibers, Cohomology, and Tangent Spaces

Projection $\pi: I \to V^{2,1}$, $\pi(x, y) = x$, makes the incidence variety into the total space of a vector bundle whose fiber at $x$ is the set of $y$ such that $x^T q y = 0$ for all $q \in Q^{\mathrm{sym}}$. This fiber coincides with the space of 2-cocycles $Z^2(\mu_x)$ in the relevant cohomology theory:

• Hochschild 2-cocycles when $x$ is associative.
• Harrison 2-cocycles for commutative $x$.
• Leibniz or Chevalley–Eilenberg cocycles for Leibniz or Lie algebra structures, respectively.

At each $x$, the tangent space of the law variety matches the subspace of 2-cocycles, and the tangent to the $\mathrm{GL}(V)$-orbit is the space of 2-coboundaries, making the stack tangent at $[\mu_x]$ naturally $H^2(\mu_x)$ (Kaygun, 21 Nov 2025).

In the representation-theoretic setting of principal series representations, integration over open orbits yields canonical $\mathrm{GL}(n)$-equivariant bilinear pairings, where uniqueness is ensured by the one-dimensionality of the $\mathrm{Hom}$ spaces of $\mathrm{GL}(n)$-equivariant functionals (Li et al., 2021).

5. Equivariance, Bilinear Pairings, and Normalization

GL-equivariance governs both the geometric and analytic properties of incidence varieties and their applications:

• In moduli and deformation theory, the action ensures orbits in the space of algebra laws correspond to isomorphism classes of algebra structures.
• For Rankin–Selberg integrals, both the standard and open-orbit incarnations yield $\mathrm{GL}(n)$-equivariant bilinear forms. Equivariance is exhibited by the transformation property $$A(s; g \cdot f, g \cdot f') = |\det g|^s A(s; f, f')$$ for $g \in \mathrm{GL}(n)$.

Normalization constants relating different incarnations (e.g., Rankin–Selberg integral $Z$ and orbit integral $A$) appear as explicit products of local Tate $\gamma$-factors and signs: $$c(\nu, \nu'; s) = \mathrm{sgn}(\nu; \nu') \prod_{i+j < n} \gamma(s; \nu_i \nu_j')$$ (Li et al., 2021).

6. Open Strata, Discriminants, and Rigidity

Open loci of "maximally nondegenerate" algebra structures are defined using invariant bilinear forms:

• Associative/commutative: The principal trace form $T_x(a, b) = \mathrm{Tr}(L_{a b})$ with discriminant $\Delta(x) = \det D_x$. Nonvanishing discriminant characterizes semisimple algebras, and on these open sets, all higher cohomologies vanish, rendering the orbits rigorous and isolated in the coarse moduli space.
• Leibniz and Lie: The right-Killing form $\kappa_R(a, b) = \mathrm{Tr}(R_a R_b)$ and its determinant characterize semisimple Lie brackets. Where nondegenerate, the cohomological obstructions vanish, and open GL-orbits correspond to rigid points (Kaygun, 21 Nov 2025).

The open-orbit geometry ensures that integration over these orbits naturally links local period integrals, explicit Euler factors, and recurrence relations in L-functions (Li et al., 2021).

7. Analytic Applications and Geometric Implications

Incidence-variety geometry informs the analytic continuation and convergence properties of integrals arising in representation theory. As the unique open GL-orbit is open and dense, any integral over this orbit converges absolutely on a suitable vertical strip in $s$ and meromorphically continues after normalization by appropriate L-factors. This underlies the functional equations and analytic behavior of Rankin–Selberg L-functions.

A plausible implication is that GL-equivariant bilinear incidence varieties function as a universal geometric model, unifying orbit theory, deformation theory, canonical pairings, and analytic constructions in both algebraic geometry and automorphic representation theory (Li et al., 2021, Kaygun, 21 Nov 2025).