## Algebra & Number Theory

### The Tannakian formalism and the Langlands conjectures

#### Abstract

Let $H$ be a connected reductive group over an algebraically closed field of characteristic zero, and let $Γ$ be an abstract group. In this note, we show that every homomorphism of Grothendieck semirings $ϕ:K0+[H]→K0+[Γ]$, which maps irreducible representations to irreducible, comes from a group homomorphism $ρ:Γ→H(K)$. We also connect this result with the Langlands conjectures.

#### Article information

Source
Algebra Number Theory, Volume 8, Number 1 (2014), 243-256.

Dates
Revised: 20 August 2013
Accepted: 19 September 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730140

Digital Object Identifier
doi:10.2140/ant.2014.8.243

Mathematical Reviews number (MathSciNet)
MR3207584

Zentralblatt MATH identifier
06322077

#### Citation

Kazhdan, David; Larsen, Michael; Varshavsky, Yakov. The Tannakian formalism and the Langlands conjectures. Algebra Number Theory 8 (2014), no. 1, 243--256. doi:10.2140/ant.2014.8.243. https://projecteuclid.org/euclid.ant/1513730140

#### References

• A. Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics 126, Springer, New York, 1991.
• C. Chevalley, Théorie des groupes de Lie, III: Théorèmes généraux sur les algèbres de Lie, Actualités Sci. Ind. 1226, Hermann & Cie, Paris, 1955.
• P. Deligne and J. S. Milne, “Tannakian categories”, pp. 101–228 in Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics 900, Springer, Berlin, 1982.
• D. Handelman, “Representation rings as invariants for compact groups and limit ratio theorems for them”, Internat. J. Math. 4:1 (1993), 59–88.
• D. Kazhdan and Y. Varshavsky, “On the cohomology of the moduli spaces of $F$-bundles: stable cuspidal Deligne–Lusztig part”, In preparation.
• S. Kumar, “Proof of the Parthasarathy–Ranga Rao–Varadarajan conjecture”, Invent. Math. 93:1 (1988), 117–130.
• M. Larsen and R. Pink, “Determining representations from invariant dimensions”, Invent. Math. 102:2 (1990), 377–398.
• J. R. McMullen, “On the dual object of a compact connected group”, Math. Z. 185:4 (1984), 539–552.
• R. Steinberg, “Regular elements of semisimple algebraic groups”, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49–80.