Hall Induction for Cotangent Representations

• Hall induction for cotangent representations is a mechanism that constructs cohomological and K-theoretic invariants of cotangent spaces via Hall algebras and vanishing cycles.
• It employs functorial maps and induction diagrams to relate cohomology spaces, ensuring torsion-free Borel–Moore homology and rigorous structural properties.
• The framework generalizes results from quiver representations to reductive groups, offering practical insights in geometric representation theory and equivariant K-theory.

Hall induction for cotangent representations describes a functorial mechanism by which cohomological and K-theoretic invariants of cotangent spaces to representations of reductive groups are constructed and studied via the formalism of Hall algebras. The subject is rooted in advancements of the cohomological Hall algebra (CoHA) program. It incorporates vanishing cycles, geometric representation theory, and diagrammatic induction techniques to illuminate structural properties such as torsion-freeness in Borel–Moore homology and the emergence of wheel conditions in equivariant K-theory. These methods generalize familiar patterns from quiver representations to arbitrary cotangent representations of reductive groups, revealing deep mathematical structures, symmetries, and divisibility properties (Gubarevich, 31 Dec 2025).

1. Structural Foundations: Hall Algebras, CoHA, and Vanishing Cycles

Let $G$ denote a complex reductive group and %%%%1%%%% a finite-dimensional $G$-module. The cotangent stack $T^*(V/G)$ is realized as $(T^*V \times^{g^*} \{0\})/G$, and its classical truncation is the zero fiber $\mu_V^{-1}(0) / G$ of the moment map $\mu_V: T^*V \to g^*$. On the product $T^*V \times g$, the $G$-invariant function $f(x, x^*, \xi) = \langle x^*, \xi \cdot x \rangle$ is considered; its critical locus contains $\mu_V^{-1}(0) \times g$.

For each cocharacter $\lambda \in X_*(T)$ (with $T \subset G$ a maximal torus and $W$ the Weyl group), one constructs the compactly supported vanishing-cycle cohomology complex

$$\mathcal H_{V,f,\lambda} := H^\bullet_{c,\,L_\lambda}\left(V^\lambda, \phi_{f_\lambda}\right)^\vee \left[-\dim(V^\lambda/L_\lambda)\right]$$

where $L_\lambda \subset G$ is the Levi component of a parabolic $P_\lambda$ associated to $\lambda$. The direct sum over all $\lambda$ acquires a CoHA structure, whose multiplication is governed by pull-push operations along an explicit induction diagram relating partial fixed loci and quotient stacks, following the approaches of Kontsevich–Soibelman and Davison.

2. Definition and Properties of the Hall Induction Functor

Given two cocharacters $\lambda \preceq \nu$—meaning $V^\lambda \subset V^\nu$, $l_\lambda \subset l_\nu$, $P_\lambda \subset P_\nu$, and $L_\lambda \subset L_\nu$—the CoHA multiplication yields an associative map:

$\Ind_\nu^\lambda:\; \mathcal H_{V,f,\lambda} \to \mathcal H_{V,f,\nu}.$

In the case $f=0$, this corresponds to a shuffle product between rings of Weyl-group invariants:

$$\mathbb{C}[t]^{W_\lambda} \to \mathbb{C}[t]^{W_\nu}, \quad f(t) \mapsto \sum_{\sigma \in W_\nu/W_{\lambda}} \sigma(f \cdot k_{\lambda,\nu}),$$

with kernel

$$k_{\lambda,\nu} = \frac{\prod_{\alpha:\,\langle\lambda, \alpha\rangle > 0} \alpha^{\dim V^\nu_\alpha}}{\prod_{\alpha:\,\langle\lambda, \alpha\rangle > 0} \alpha^{\dim l_{\nu,\alpha}}} \in \mathbb{C}[X^*(T)](-2\dim \pi^\nu_\lambda + 2 \dim q^\nu_\lambda).$$

The associativity extends to the induced maps between cohomology spaces, reflective of the functoriality inherent in their geometric construction.

3. Geometric Realization for Cotangent Representations

Specializing to $V \mapsto T^*V \times g$ and $f(x, x^*, \xi) = \langle x^*, \xi \cdot x \rangle$, equivariant dimensional reduction gives an isomorphism

$$\mathcal H_{T^*V \times g, f, \lambda} \simeq H^{BM}_{L_\lambda}\left(\mu_\lambda^{-1}(0)\right)[d_\lambda + 2l_\lambda]$$

where $d_\lambda = \dim T^*V^\lambda$ and $l_\lambda = \dim l_\lambda$. Hall induction therefore becomes

$\Ind_\nu^\lambda: H_{L_\lambda}^{BM}\left(\mu_\lambda^{-1}(0)\right)[d_\lambda + 2l_\lambda] \to H_{L_\nu}^{BM}\left(\mu_\nu^{-1}(0)\right)[d_\nu + 2l_\nu]$

with associativity ensured by dimensional reduction and vanishing-cycle functoriality.

4. Torsion-Freeness Theorem in Borel–Moore Homology

Consider $H^{BM}_{G \times T_s}(\mu_V^{-1}(0))$ as a module over $H_{G \times T_s}(pt)$, for $T_s$ an auxiliary torus acting via subtori of weights $(1, -1, 0)$ and $(1, 0, -1)$. The central result is:

Theorem (Gubarevich):

$H^{BM}_{G \times T_s}(\mu_V^{-1}(0))$ is torsion free as an $H_{G \times T_s}$-module; equivalently, the restriction map

$$j^*: H^{BM}_{G \times T_s}(\mu_V^{-1}(0)) \to H^{BM}_{G \times T_s}(pt) \simeq H_{G \times T_s}$$

is injective.

Sketch of Proof:

• Localization along a subtorus and application of Atiyah–Bott localization reveal $j^*$ becomes an isomorphism after inverting appropriate weights.
• By equivariant dimensional reduction, $H^{BM}_{G \times T_s}(\mu_V^{-1}(0))$ identifies with $H^{BM}_{G \times T_s}(\{(x, a): a\cdot x = 0\})$, reducible further to the nilpotent cone $\mathcal N \subset g$.
• Stratifying $\mathcal N$ via conjugacy classes, one finds even cohomological degree purity, so long exact sequences associated to the inclusions split, and $H^{BM}_{G \times T_s}(\mathcal N)$ injects into the sum over fixed points, thus establishing torsion-freeness (Gubarevich, 31 Dec 2025).

5. Wheel Conditions in Equivariant K-Theory

Extending results analogous to the KHA of the one-loop quiver, where Zhao established that the image of

$$K_{GL_n \times (\mathbb{C}^*)^2}(Comm_n) \to K_{(\mathbb{C}^*)^2} \cong \mathbb{Z}[q^{\pm1}, q'^{\pm1}][z_1^{\pm1},...,z_n^{\pm1}]^{S_n}$$

lies in the intersection of the "wheel ideals" $(1-q^{-1}z_j/z_i, 1-q'^{-1}z_k/z_j)$, the general result is:

Theorem (Gubarevich):

If $\mu_V^{-1}(0)^{G \times T_s} = \{\mathrm{pt}\}$, then the restriction

$$j^*: K_{G \times T_s}\left(\mu_V^{-1}(0)\right) \to K_{G \times T_s}(pt) \simeq K_{T \times T_s}(pt)^W$$

lands in the $W$-symmetric part of the intersection of all rank-two ideals

$$\bigcap_{(l \subset V,\, l' \subset V^*)} (1 - \chi_l^{-1},\, 1 - \chi_{l'}^{-1}),$$

where $\chi_l$, $\chi_{l'}$ are characters associated to chosen coordinate lines and the intersection runs over all pairs making a certain Cartesian square. The proof reduces the calculation to summing over fixed-point contributions and examining their divisibility and symmetry properties.

6. Concrete Examples and Applications

a) One-Loop Quiver

For the adjoint action of $GL_n$ on $\mathfrak{g}$ with $T_s = (\mathbb{C}^*)^2$ scaling as $(x, y) \mapsto (q x, q' y)$, the wheel condition theorem recovers Zhao's result:

$$\bigcap_{i \neq j \neq k} (1-q^{-1} z_j / z_i,\, 1 - q'^{-1}z_k / z_j) \subset \mathbb{Z}[q^{\pm1}, q'^{\pm1}][z_1^{\pm1},\dots, z_n^{\pm1}]^{S_n}$$

b) Irreducible $SL_2$-Modules

Taking $V = \mathrm{Sym}^n(\mathbb{C}^2)$, there are three families of coordinate-line pairs for $k + l = n$:

$$e_1^k e_2^l \cup (e_1^*)^{k \pm 1} (e_2^*)^{l \mp 1},\qquad e_1^k e_2^l \cup (e_1^*)^k(e_2^*)^l,$$

yielding wheel ideals such as

$$(1 - z^{-(k-l)} q_1^{-k}q_2^{-l},\, 1 - z^{k-l\pm2} q_1^{k\pm1} q_2^{l\mp1}),\quad (1 - z^{-(k-l)} q_1^{-k}q_2^{-l},\, 1 - z^{k-l} q_1^k q_2^l),$$

summed and symmetrized under the Weyl group $S_2$.

Together, these examples demonstrate that the CoHA-induced Hall induction processes apply broadly to cotangent representations, yield torsion-free equivariant Borel–Moore homology, produce natural shuffle-type multiplication formulas, and encode precise divisibility constraints—wheel conditions—in equivariant K-theory (Gubarevich, 31 Dec 2025).