science

Kirillov model

Kirillov model is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Kirillov model rather than just read about it. In short: In mathematics, the Kirillov model, studied by Kirillov (1963), is a realization of a representation of GL2 over a local field on a space of functions on the local field. If G is the algebraic group GL2 and F is a non-Archimedean local field, and τ is a fixed nontrivial character of the additive group of F and π is an irreducible representation of G(F), then the Kirillov model for π is a representation π on a space…

Key takeaways

Kirillov model belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Kirillov model to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Kirillov model from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Kirillov model, studied by Kirillov (1963), is a realization of a representation of GL2 over a local field on a space of functions on the local field. If G is the algebraic group GL2 and F is a non-Archimedean local field, and τ is a fixed nontrivial character of the additive group of F and π is an irreducible representation of G(F), then the Kirillov model for π is a representation π on a space of locally constant functions f on F* with compact support in F such that

π ( ( a b 0 1 ) ) f ( x ) = τ ( b x ) f ( a x ) . {\displaystyle \pi \left({\begin{pmatrix}a&b\\0&1\end{pmatrix}}\right)f(x)=\tau (bx)f(ax).}

Jacquet & Langlands (1970) showed that an irreducible representation of dimension greater than 1 has an essentially unique Kirillov model. Over a local field, the space of functions with compact support in F* has codimension 0, 1, or 2 in the Kirillov model, depending on whether the irreducible representation is cuspidal, special, or principal. The Whittaker model can be constructed from the Kirillov model, by defining the image Wξ of a vector ξ of the Kirillov model by

Wξ(g) = π(g)ξ(1) where π(g) is the image of g in the Kirillov model. Bernstein (1984) defined the Kirillov model for the general linear group GLn using the mirabolic subgroup. More precisely, a Kirillov model for a representation of the general linear group is an embedding of it in the representation of the mirabolic group induced from a non-degenerate character of the group of upper triangular matrices.

References Bernstein, Joseph N. (1984), "P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-Archimedean case)", Lie group representations, II (College Park, Md., 1982/1983), Lecture Notes in Math., vol. 1041, Berlin, New York: Springer-Verlag, pp. 50–102, doi:10.1007/BFb0073145, ISBN 978-3-540-12715-4, MR 0748505 Kirillov, A. A. (1963), "Infinite-dimensional unitary representations of a second-order matrix group with elements in a locally compact field", Doklady Akademii Nauk SSSR, 150: 740–743, ISSN 0002-3264, MR 0151552 Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114, vol. 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3-540-04903-6, MR 0401654, S2CID 122773458

Worked examples

Example 1 — a first encounter with Kirillov model

Start with the simplest possible case. Write down what Kirillov model claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Kirillov model before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Kirillov model ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Kirillov model

In research: Kirillov model appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Kirillov model in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Kirillov model is common in secondary-school and first-year university syllabi. It links to neighbouring topics Automorphic forms, Langlands program, Representation theory, so understanding it makes those chapters shorter.
In everyday life: Look for Kirillov model outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Kirillov model in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kirillov model means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Kirillov model out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kirillov model in simple terms?

Why does Kirillov model matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Kirillov model?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Kirillov model.