Higher Verlinde categories of reductive groups

We define tensor categories ${\sf Ver}{p^n}(G)$ in characteristic $p$ for connected reductive groups $G$ and positive integers $n$, generalising the semisimple Verlinde categories ${\sf Ver}p(G)$ originating from Gelfand-Kazhdan and the higher Verlinde categories ${\sf Ver}{p^n}$ for ${\rm SL}2$ defined by Benson-Etingof-Ostrik. The construction is based on the definition of ${\sf Ver}{p^n}$ as an abelian envelope of a quotient of a category of tilting modules, but we also introduce an expanded construction which refines the ${\rm SL}2$ case and gives new results. In particular, the union ${\sf Ver}{p^\infty}(G)$ can be derived from the perfection of $G$; certain exact sequences in ${\sf Rep}G$ map to exact sequences in ${\sf Ver}{p^n}(G)$; and the underlying abelian category of ${\sf Ver}_{p^n}$ can be expressed as a subcategory of ${\sf Rep}{\rm SL}_2$, or as a Serre quotient of a subcategory of ${\sf Rep}{\rm SL}_2$.

💡 Research Summary

In this paper the author introduces a family of tensor categories ${\sf Ver}_{p^n}(G)$ for any connected reductive algebraic group $G$ over an algebraically closed field $k$ of characteristic $p>0$ and any positive integer $n$. The construction simultaneously generalises two previously known families: (1) the semisimple Verlinde categories ${\sf Ver}p(G)$ introduced by Gelfand‑Kazhdan, which are obtained from the category of tilting modules of $G$ by quotienting out the ideal of negligible morphisms, and (2) the higher Verlinde categories ${\sf Ver}{p^n}$ defined by Benson‑Etingof‑Ostrik for $SL_2$, which arise as the abelian envelope of a quotient of the tilting category of $SL_2$ by a different tensor ideal $I_n$.

The key technical tool is a theorem of Steinberg (re‑formulated in recent work of Coulembier‑Etingof‑Ostrik) guaranteeing the existence of an abelian envelope for any pseudo‑tensor category equipped with a minimal tensor ideal. Using this, the author first defines ${\sf Ver}_{p^n}(G)$ as the abelian envelope of the quotient $Tilt,G/F^{-1}(I_n(SL_2))$, where $F: Tilt,G\to Tilt,SL_2$ is induced by a fixed principal homomorphism $\varphi:SL_2\to G$.

To obtain a more flexible description, the author enlarges the source category from $Tilt,G$ to a full subcategory $T_n\subset Rep,G$ consisting of all objects $X$ such that $X\otimes St_{n-1}$ is a tilting module (here $St_{n-1}$ is the $(n-1)$‑st Steinberg module). Inside $T_n$ a tensor ideal $I_n$ is defined analogously to $I_n(SL_2)$. The abelian envelope of $T_n/I_n$ is shown to coincide with the previously defined ${\sf Ver}_{p^n}(G)$. This “expanded construction’’ makes the theory work uniformly for any $G$ and provides a more efficient proof of many properties that were originally established only for $SL_2$.

The main theorem (Theorem 1) lists eight fundamental properties of ${\sf Ver}_{p^n}(G)$ under the standing assumptions $p\ge h$ (where $h$ is the Coxeter number of $G$) and the validity of Donkin’s tensor product theorem (known for $p\ge 2h-4$). Highlights include:

1. Finite tensor category – the indecomposable projective objects are precisely the images of tilting modules $T(\lambda)$ with highest weights lying in a specific region determined by the fundamental alcove $A$, the $p^{n-1}$‑restricted weight set $\Lambda_{n-1}$ and the shift $(p^n-1-1)\rho$.

2. Abelian envelope description – ${\sf Ver}_{p^n}(G)$ is the abelian envelope of $T_n/I_n$, where $T_n$ is defined via the Steinberg module condition.

3. Simple objects and Steinberg tensor product – simples are the images of $L(\lambda)$ for $\lambda$ in $(\Lambda_{n-1}+p^{n-1}A)\cap X(T)$. Moreover, writing $\lambda=\lambda_0+p\lambda_1+\dots+p^{n-1}\lambda_{n-1}$ with each $\lambda_i$ $p$‑restricted, one has $L(\lambda)=L(\lambda_0)\otimes L(p\lambda_1)\otimes\cdots\otimes L(p^{n-1}\lambda_{n-1})$ inside ${\sf Ver}_{p^n}(G)$.

4. Compatibility with the principal $SL_2$ map – the restriction functor $T_n(G)\to T_n(SL_2)$ induces a tensor functor ${\sf Ver}{p^n}(G)\to{\sf Ver}{p^n}$ commuting with the principal embedding.

5. Frobenius twist inclusions – there are natural fully faithful tensor functors ${\sf Ver}{p^n}(G)\hookrightarrow{\sf Ver}{p^{n+1}}(G)$ induced by the Frobenius twist $X\mapsto X^{(1)}$ on $T_n$.

6. Central decomposition – when $p>h$, the category splits as a Deligne product ${\sf Ver}_{p^n}(G/Z)\boxtimes\operatorname{sVec}(Z,z)$, where $Z$ is the centre of $G$ and $z$ encodes the parity of the central character.

7. Infinite union – the direct limit ${\sf Ver}{p^\infty}(G)=\bigcup{n}{\sf Ver}{p^n}(G)$ is the abelian envelope of $T\infty/I_\infty$, where $T_\infty$ lives inside the representation category of the perfection $G_{\mathrm{perf}}$ and $I_\infty$ is the corresponding tensor ideal.

8. Exactness preservation – the canonical functor $T_n\to{\sf Ver}_{p^n}(G)$ sends bounded exact sequences to exact sequences, ensuring that homological information is retained.

In the special case $G=SL_2$, the author gives a concrete description of the underlying abelian category of ${\sf Ver}_{p^n}$. Three subcategories of $Rep,SL_2$ are introduced:

• $A_n$: objects whose highest weights are $<p^n-1$;
• $B_n$: the Serre subcategory of $A_n$ generated by simples $L_i$ with $(p-1)p^{n-1}\le i<p^n-1$;
• $C_n$: objects in $A_n$ having no non‑zero morphisms to or from any object of $B_n$.

The author proves $C_n\cong A_n/B_n\cong{\sf Ver}{p^n}$, thereby identifying the higher Verlinde category with a Serre quotient of a natural subcategory of $Rep,SL_2$. This recovers the known description for $n=1$ and extends it to all $n\ge2$, providing the first explicit non‑semisimple examples of tensor categories fibering over ${\sf Ver}{p^n}$.

The paper concludes with several avenues for future work: interpreting ${\sf Ver}_{p^\infty}(G)$ via Tannakian or “Tannakian‑like’’ formalism to identify the affine group scheme whose representation category it models; extending the construction to quantum groups at roots of unity for arbitrary $G$ (generalising results of