Hecke algebra of a pair¶
An algebra built from supported double-coset-invariant functions or sums for an admissible group-subgroup pair, with multiplication by convolution.
Core Idea¶
A Hecke algebra of a pair turns a suitable group G and subgroup K or P into an algebra whose elements are supported sums or functions constant on the subgroup's double cosets. It combines those elements by convolution, or by the corresponding multiplication in a finite group algebra. The subgroup chooses the double-coset classes; the support and coefficient rules determine which combinations and products are valid.[ref-044471a8c4b0][ref-c05ae1832200]
This is a kind of Associative Algebra. Its extra identity is the pair-specific, double-coset-invariant construction. The term does not name one ring for every abstract pair: Curtis and colleagues work with a finite characteristic-zero algebra, while Gross works with an integer-valued spherical function ring on a p-adic group.[ref-044471a8c4b0][ref-c05ae1832200]
Use Within the Domain¶
In representation theory and harmonic analysis, the pair algebra lets a calculation on group elements be expressed through subgroup double cosets. Before using it, state the pair, which functions or sums are admitted, their coefficient base, and how multiplication is defined. Those choices decide what the algebra actually is.[ref-044471a8c4b0][ref-c05ae1832200]
The live Associative Algebra parent supplies the scalar-module carrier and bilinear associative product. The Hecke construction adds a subgroup and its two-sided invariance. Gross's commutativity result applies to his spherical pair; commutativity is not required of every pair algebra.[ref-044471a8c4b0][ref-c05ae1832200]
Clarity¶
Keep four parts separate: an admissible group-subgroup pair, a double-coset-invariant carrier, a support and coefficient convention, and a closed associative product. A list of double cosets alone supplies an indexing set, not an algebra. A whole group algebra lacks the chosen subgroup's bi-invariance rule.[ref-044471a8c4b0][ref-c05ae1832200]
The finite and p-adic versions use different bookkeeping. For finite G, the normalized subgroup sum e selects a corner e(kG)e, with multiplication inherited from kG. For Gross's compact subgroup K, functions have compact support and integral values, while normalized Haar convolution supplies the product and char(K) the unit.[ref-044471a8c4b0][ref-c05ae1832200]
Manages Complexity¶
Double-coset invariance replaces many group-element terms with basis elements indexed by P\G/P or K\G/K. Products can then be studied through the coefficients of these basis elements. This compression is useful only after the support and multiplication assumptions have been verified; it does not by itself prove commutativity or a representation theorem.[ref-044471a8c4b0][ref-c05ae1832200]
Abstract Reasoning¶
To test a proposed example, identify the subgroup and its double cosets, then check that the chosen supported sums or functions are closed under the stated product. Verify the coefficient base and whether the product has a unit. If bi-invariance or closure fails, the proposed object is not established as this pair Hecke algebra.[ref-044471a8c4b0][ref-c05ae1832200]
The two source constructions show why conclusions must retain their hypotheses. A fact proved for Gross's spherical Z-ring does not automatically hold for Curtis's characteristic-zero corner merely because both have double-coset bases.[ref-044471a8c4b0][ref-c05ae1832200]
Knowledge Transfer¶
The four-role test transfers between finite and locally compact group settings: choose a pair, select double-coset-invariant supported elements, fix coefficients, and verify associative multiplication. The details of finite averaging and Haar integration do not transfer unchanged. Outside group-subgroup mathematics, a convolution-like operation alone is not this named construction.[ref-044471a8c4b0][ref-c05ae1832200]
Example¶
Finite parabolic pair. Curtis and colleagues take a finite group G with a (B,N)-pair and a parabolic subgroup P. Over a characteristic-zero field k, the normalized P-sum e yields the corner e(kG)e. Its P-double-coset sums give the carrier basis; finite k-linear combinations supply support and coefficients; multiplication comes from kG. The unit is e.[^ref-044471a8c4b0]
Spherical p-adic pair. Gross takes a split reductive G(F) with hyperspecial compact K=G(A). The carrier is the locally constant, compactly supported, integer-valued K-bi-invariant functions, with K-double-coset characteristic functions as a basis. The coefficient base is Z; Haar measure is normalized by vol(K)=1; convolution is the product and char(K) its unit. This special algebra is commutative, but that feature is not a universal membership test.[^ref-c05ae1832200]
Relationships to Other Abstractions¶
Current abstraction Hecke algebra of a pair Domain-specific
Parents (1) — more general patterns this builds on
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Hecke algebra of a pair is a kind of Associative algebra Domain-specific
Every admissible pair Hecke algebra is an associative algebra over its declared commutative coefficient base; the subgroup double-coset construction supplies its narrower identity.
Hierarchy path (1) — routes to 1 parentless root
- Hecke algebra of a pair → Associative algebra → Algebra over a Ring → Linearity
Neighborhood in Abstraction Space¶
Hecke algebra of a pair sits in a sparse region of the domain-specific corpus (83rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2721 abstractions)
Nearest neighbors
- Projective representation — 0.83
- Ahlswede–Daykin inequality — 0.82
- J-homomorphism — 0.82
- Hochschild homology — 0.81
- Transfer (group theory) — 0.81
Computed from structural-signature embeddings · 2026-10-17
Not to Be Confused With¶
A Hecke character is a different analytic object. A group algebra includes elements without imposing this pair's double-coset invariance. A bare double-coset set lacks the supported algebra carrier and multiplication. The live Convolution Prime describes fixed-kernel filtering, not the full signature of every pair Hecke algebra. The finite e(kG)e and Gross's integer-valued ring should not be collapsed into one coefficient or measure convention.[ref-044471a8c4b0][ref-c05ae1832200]
References¶
[^ref-044471a8c4b0]: Charles W. Curtis, Nagoyashi Iwahori, and Robert Kilmoyer, “Hecke Algebras and Characters of Parabolic Type of Finite Groups with (B,N)-Pairs,” Publications mathématiques de l'I.H.É.S. 40 (1971): 81–116. Introduction and Definition 1.3, printed pp. 81 and 83; archive PDF pp. 2 and 4. https://www.numdam.org/item/PMIHES_1971__40__81_0.pdf [^ref-c05ae1832200]: Benedict H. Gross, “On the Satake Isomorphism,” in Galois Representations in Arithmetic Algebraic Geometry, London Mathematical Society Lecture Note Series 254 (1998): 223–238. Author-hosted preprint, section 2, PDF pp. 3–5, especially equations (2.1), (2.5), (2.7) and Proposition 2.10. https://people.math.harvard.edu/~gross/preprints/sat.pdf