Harish-Chandra

How to use this course

Fourier analysis decomposes a function on the circle into the characters \(e^{in\theta}\). Harish-Chandra spent thirty-five years doing the same for functions on a semisimple Lie group such as \(\mathrm{SL}(n,\mathbb R)\). The building blocks are now infinite-dimensional unitary representations, and the problem is to find them, compute their characters, and write down the measure that reassembles a function from its pieces. He solved this problem for real reductive groups, set up the corresponding theory for \(p\)-adic groups, and laid the analytic foundations on which Robert Langlands built his programme.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The group \(\mathrm{SL}(2,\mathbb R)\) recurs throughout as a test case: every general theorem is worked out for it. Many statements are given for connected semisimple groups with finite centre; the references give the general hypotheses.

Timeline

Kanpur and Allahabad, 1923–1943

Harish-Chandra was born on 11 October 1923 in Kanpur, in what was then the United Provinces of British India. His father, Chandrakishore, was a civil engineer who inspected irrigation canals. His mother, Satyagati Seth, was the daughter of a lawyer, Ram Sanehi Seth, and the boy spent much of his childhood in his maternal grandfather’s house in Kanpur.

He moved through school quickly. He finished Christ Church High School at fourteen and intermediate college at sixteen. At the University of Allahabad he took a B.Sc. in 1941 and an M.Sc. in physics in 1943, placing first in the province. Two things shaped him there. The physicist K. S. Krishnan, his teacher, encouraged him toward theoretical physics. And he read Paul Dirac’s Principles of Quantum Mechanics, the book that decided his first career.

Bangalore and Cambridge, 1943–1947

In 1943 Harish-Chandra went to the Indian Institute of Science in Bangalore to work as a research fellow with Homi Bhabha, who had returned to India during the war. He wrote papers on relativistic quantum theory, two of them jointly with Bhabha, extending results of Dirac. His first papers appeared in the Proceedings of the Royal Society in 1944. Bhabha asked Dirac to check the proofs, and this led Dirac to accept Harish-Chandra as a research student.

In 1945 he entered Gonville and Caius College, Cambridge, as Dirac’s research student. He saw his supervisor rarely and found Dirac’s lectures close to the book. He attended lectures by J. E. Littlewood and Philip Hall instead and began to read mathematics seriously. At a lecture by Wolfgang Pauli he pointed out a mistake; the two became friends.

Dirac suggested the topic that defined his life: the infinite-dimensional irreducible representations of the Lorentz group. Finite-dimensional representations of the Lorentz group describe spinors and tensors; relativistic wave equations with infinitely many components need infinite-dimensional ones. Harish-Chandra classified the irreducible representations of the Lie algebra of the Lorentz group and identified the unitary ones. He received his Ph.D. in 1947. In the same year Valentine Bargmann (for \(\mathrm{SL}(2,\mathbb R)\)) and Israel Gelfand with Mark Naimark (for the Lorentz group) published rigorous treatments of closely related problems. Harish-Chandra later judged his own thesis to have the formal style of a physicist’s paper, with delicate analytic points passed over. The judgement pushed him toward mathematics.

Princeton and the turn to mathematics, 1947–1950

Dirac spent 1947–48 at the Institute for Advanced Study in Princeton and took Harish-Chandra with him as his assistant. Princeton changed his direction. He met Hermann Weyl, whose work on compact groups was the model for everything he later did, along with Emil Artin and Claude Chevalley, whose algebraic treatment of Lie theory he absorbed. Dirac told him that he cared only for what nature does, and not for proofs. Harish-Chandra concluded that he lacked the physicist’s intuition and was a mathematician.

His last physics papers appeared in 1948 and his first mathematics papers in 1949. He stayed a second year in Princeton, then spent 1949–50 at Harvard as a Jewett Fellow, nominally to learn algebraic geometry from Oscar Zariski. By 1950 he had chosen his subject: the representation theory of all semisimple Lie groups, approached through their Lie algebras and through analysis, with the aim of a Plancherel formula as complete as Weyl’s theory for compact groups.

Representation theory of non-compact groups was then new. Bargmann, Gelfand and Naimark had treated a few groups by hand. Harish-Chandra’s plan was to find general principles. He worked alone for most of his life and published only one joint paper in mathematics.

SL(2,ℝ): the model case

For the general reader. The group \(\mathrm{SL}(2,\mathbb R)\) of real \(2\times2\) matrices of determinant 1 acts on the hyperbolic plane. It is the simplest group whose interesting unitary representations are all infinite-dimensional. Every idea in Harish-Chandra’s work can be seen here first, and his thesis began with its close relative, the Lorentz group \(\mathrm{SL}(2,\mathbb C)\).

Lie algebra and \(K\)-types

Let \(K=\mathrm{SO}(2)\), with characters \(k_\theta\mapsto e^{in\theta}\), \(n\in\mathbb Z\). In the complexified Lie algebra \(\mathfrak g_{\mathbb C}=\mathfrak{sl}(2,\mathbb C)\) choose

so that \([h,e]=2e\), \([h,f]=-2f\), \([e,f]=h\), and \(h\) spans the complexified Lie algebra of \(K\). If \(v_n\) is a vector of \(K\)-type \(n\), meaning \(h v_n=n v_n\), then \(e v_n\) has type \(n+2\) and \(f v_n\) has type \(n-2\). The Casimir element

commutes with everything and acts by a scalar on an irreducible module. We write that scalar as \((t^2-1)/2\); the number \(t\), defined up to sign, is the infinitesimal character (Lesson 05).

From representations to modules

Given a representation \(\pi\) on a Hilbert space \(V\), the \(K\)-finite vectors \(V_K\) form a dense subspace on which \(\mathfrak g\) acts. It is a \((\mathfrak g,K)\)-module: compatible actions of the Lie algebra and of \(K\). This algebraic shadow is Harish-Chandra’s key device. For \(\mathrm{SL}(2,\mathbb R)\) each \(K\)-type of an irreducible module occurs at most once, all types have the same parity, and the module is determined by its set of \(K\)-types and by \(t\).

The principal series \(P^{\pm,s}\) acts on functions on \(\mathbb R^2\setminus\{0\}\) satisfying \(F(cx)=|c|^{s-1}\operatorname{sgn}(c)^{\varepsilon}F(x)\), by \(F\mapsto F\circ g^{-1}\). Restricting to the unit circle shows that its \(K\)-types are all \(n\equiv\varepsilon \pmod 2\), and its infinitesimal character is \(t=s\). The discrete series \(D_k^{\pm}\) can be realized on holomorphic (or antiholomorphic) functions on the upper half-plane, square-integrable for the measure \(y^{k-2}\,dx\,dy\). Its matrix coefficients are square-integrable on \(G\): that is the meaning of “discrete series”.

The enveloping algebra and its centre

For the general reader. On the circle, the derivative \(d/d\theta\) acts on \(e^{in\theta}\) by the number \(in\), which labels the representation. For a semisimple group the role of \(d/d\theta\) is played by a commutative algebra of invariant differential operators, and each irreducible representation gives a list of eigenvalues. Harish-Chandra’s 1951 paper computed this algebra exactly. The answer, his isomorphism, is used across representation theory and beyond.

The theorem

Let \(\mathfrak g\) be a complex semisimple Lie algebra, \(\mathfrak h\) a Cartan subalgebra, \(\Phi^+\) a choice of positive roots, \(\rho=\tfrac12\sum_{\alpha\in\Phi^+}\alpha\), \(W\) the Weyl group, and \(\mathfrak n^{\pm}\) the sums of positive and negative root spaces. By the Poincaré–Birkhoff–Witt theorem, \(U(\mathfrak g)=U(\mathfrak h)\oplus(\mathfrak n^-U(\mathfrak g)+U(\mathfrak g)\mathfrak n^+)\). Let \(p\) be the projection onto \(U(\mathfrak h)=S(\mathfrak h)\), viewed as polynomial functions on \(\mathfrak h^*\).

Every \(z\in Z(\mathfrak g)\) acts on the Verma module of highest weight \(\lambda\) by the scalar \(\chi_\lambda(z)=\gamma(z)(\lambda+\rho)\). Hence every irreducible module with a highest weight, and every irreducible Harish-Chandra module, has an infinitesimal character \(\chi:Z(\mathfrak g)\to\mathbb C\), and:

Proof sketch. A central \(z\) has weight 0 for \(\mathfrak h\), and this forces \(z-p(z)\in U(\mathfrak g)\mathfrak n^+\). Such terms kill a highest weight vector, so \(z\) acts on the Verma module \(M(\lambda)\) by \(p(z)(\lambda)\). For dominant integral \(\lambda\), the module \(M(s_\alpha\cdot\lambda)\), with \(w\cdot\lambda=w(\lambda+\rho)-\rho\), embeds in \(M(\lambda)\), so \(p(z)(\lambda)=p(z)(s_\alpha\cdot\lambda)\). Shifting by \(\rho\) turns the dot action into the linear action, and Zariski density of dominant integral weights gives \(W\)-invariance of \(\gamma(z)\). Injectivity follows by examining leading terms. Surjectivity uses Chevalley’s restriction theorem: \(S(\mathfrak g)^{\mathfrak g}\to S(\mathfrak h)^W\) is onto, and the symmetrization map transports this to \(U(\mathfrak g)\) up to lower-order terms.

In the same paper Harish-Chandra gave an algebraic construction of the irreducible finite-dimensional representations, as quotients of what are now called Verma modules, independently of Chevalley. The infinitesimal character became his main tool for analysis: on \(G/K\), elements of \(Z(\mathfrak g)\) act as invariant differential operators, and on a group they commute with both left and right translation.

Harish-Chandra modules and characters

For the general reader. An infinite-dimensional representation has no trace in the usual sense, since the identity operator has infinite trace. Harish-Chandra showed that the trace still makes sense after smearing the group element with a smooth test function, and that the result is given by an honest function on the group. These characters determine the representation, as they do for finite groups, and they made a Fourier theory on Lie groups possible.

Let \(G\) be a connected semisimple Lie group with finite centre, \(K\) a maximal compact subgroup, and \(\mathfrak g\) the complexified Lie algebra. A representation \(\pi\) of \(G\) on a Hilbert space is admissible if each irreducible representation \(\delta\) of \(K\) occurs in \(\pi|_K\) with finite multiplicity. A Harish-Chandra module is a \((\mathfrak g,K)\)-module that is finitely generated over \(U(\mathfrak g)\) and admissible.

The character

For \(f\in C_c^\infty(G)\) put \(\pi(f)=\int_G f(g)\pi(g)\,dg\). Harish-Chandra proved that for irreducible admissible \(\pi\) the operator \(\pi(f)\) is of trace class, and that \(\Theta_\pi(f)=\operatorname{tr}\pi(f)\) is a distribution on \(G\). It is invariant under conjugation, and since \(Z(\mathfrak g)\) acts on \(\pi\) by the infinitesimal character \(\chi_\pi\), it satisfies \(z\,\Theta_\pi=\chi_\pi(z)\Theta_\pi\) for all \(z\in Z(\mathfrak g)\), acting as bi-invariant differential operators. Such a distribution is an invariant eigendistribution. Irreducible admissible representations with the same character are infinitesimally equivalent.

The proof is a long analytic argument. Harish-Chandra first proved the analogous theorem on the Lie algebra, using the Fourier transform and the fact that the eigen-equations become a holonomic system near nilpotent elements, then transferred it to the group by the exponential map and a descent to centralizers of semisimple elements.

The discrete series

For the general reader. Some representations of a group appear in the space of square-integrable functions on it as isolated pieces, the way a single frequency appears in a periodic signal. These are the discrete series. Harish-Chandra decided exactly which groups have them, found them all, and wrote down their characters, a theorem many regard as the summit of his work.

An irreducible unitary representation \(\pi\) belongs to the discrete series if it occurs as a closed subrepresentation of the regular representation on \(L^2(G)\); equivalently, its matrix coefficients \(g\mapsto\langle\pi(g)u,v\rangle\) are square-integrable. Then there is a constant \(d_\pi>0\), the formal degree, with \(\int_G|\langle\pi(g)u,v\rangle|^2dg=d_\pi^{-1}\|u\|^2\|v\|^2\), generalizing \(1/\dim\) for compact groups.

Compare Weyl’s character formula for compact groups: the numerator there is summed over the whole Weyl group \(W\). For non-compact \(G\), \(W_K\) is smaller, and \(|W/W_K|\) discrete series share each regular infinitesimal character. The formula determines \(\Theta_{\pi_\lambda}\) on \(T\); its values on non-compact Cartan subgroups are fixed by the differential equations and by a growth condition. The formal degree is proportional to \(\prod_{\alpha\in\Phi^+}|\langle\lambda,\alpha\rangle|\), as in Weyl’s dimension formula.

Strategy of proof. Harish-Chandra worked with tempered invariant eigendistributions. He showed that one is determined by its restriction to \(T\), constructed for each regular \(\lambda\) an eigendistribution \(\Theta_\lambda\) with the stated values on \(T\), using Fourier analysis on the Lie algebra and his theory of the Schwartz space \(\mathcal C(G)\), and then proved that \(\Theta_\lambda\) is the character of a square-integrable representation by showing that discrete series exhaust what remains of \(L^2(G)\) after the continuous spectrum is removed. Existence was settled without an explicit model. Geometric realizations came later, from Wilfried Schmid (on \(L^2\)-cohomology of line bundles on \(G/T\)) and from Michael Atiyah and Schmid (1977).

Spherical functions and the Plancherel formula

For the general reader. The Plancherel formula says how much of each representation is needed to rebuild an arbitrary function on the group. For the circle every frequency has equal weight. For a semisimple group the weights vary, and Harish-Chandra found that they are governed by a single function, his \(c\)-function, which records how spherical waves on a symmetric space behave far from the origin.

Spherical functions

Let \(G=KAN\) be an Iwasawa decomposition, write \(g=k\exp H(g)\,n\), and let \(\rho\) be half the sum of the positive restricted roots, counted with multiplicity. For \(\lambda\in\mathfrak a^*_{\mathbb C}\), Harish-Chandra’s formula (1958) defines

These are exactly the \(K\)-bi-invariant joint eigenfunctions of the invariant differential operators with \(\varphi(1)=1\), and \(\varphi_\lambda=\varphi_\mu\) iff \(\mu\in W\lambda\).

Harish-Chandra proved the inversion formula under assumptions about \(c\) that were then verified by Simon Gindikin and Fridrikh Karpelevich (1962), who computed \(c\) as a product of rank-one factors, each a ratio of gamma functions. In the rank-one case of the hyperbolic plane \(\mathbb H^2=\mathrm{SL}(2,\mathbb R)/\mathrm{SO}(2)\) with curvature \(-1\), with \(\rho=\tfrac12\) and \(r\) the hyperbolic distance from the origin:

The full theorem

The building blocks are the discrete series of the groups \(M\), which explains why Harish-Chandra had to settle the discrete series first. For \(\mathrm{SL}(2,\mathbb R)\) the formula has the shape

with the discrete series and the even and odd principal series as the cuspidal data, and a constant \(c>0\) fixed by the normalization of Haar measure and of \(d\nu\). The opening illustration plots these weights. The complementary series and the trivial representation do not appear: they are not tempered.

Arithmetic groups and automorphic forms

For the general reader. Modular forms are functions on the upper half-plane with symmetry under the group \(\mathrm{SL}(2,\mathbb Z)\) of integer matrices. Harish-Chandra and Armand Borel proved that the analogous quotient is finite in volume for every arithmetic group, and Harish-Chandra proved that the spaces of automorphic forms are finite-dimensional. These theorems turned modular forms into a chapter of representation theory, the setting for the Langlands programme.

The proof builds explicit fundamental domains, Siegel sets, by reduction theory in the style of Hermite, Minkowski and Siegel, and estimates their volume using the Iwasawa decomposition. The compactness criterion was proved independently by George Mostow and Tsuneo Tamagawa.

Automorphic forms

Fix such \(\Gamma\) in a semisimple \(G=G(\mathbb R)\). An automorphic form is a smooth function \(\phi\) on \(\Gamma\backslash G\) that is right \(K\)-finite, \(Z(\mathfrak g)\)-finite, and of moderate growth. For \(\mathrm{SL}(2,\mathbb R)\) and \(\Gamma=\mathrm{SL}(2,\mathbb Z)\), a modular form of weight \(k\) gives one: \(\phi(g)=(ci+d)^{-k}f(g\cdot i)\) for \(g=\begin{pmatrix}a&b\\c&d\end{pmatrix}\).

For a parabolic subgroup \(P=MN\) defined over \(\mathbb Q\), the constant term of \(\phi\) along \(P\) is \(\phi_P(g)=\int_{(\Gamma\cap N)\backslash N}\phi(ng)\,dn\). A cusp form is an automorphic form whose constant terms along all proper rational parabolics vanish. Cusp forms are rapidly decreasing and lie in the discrete spectrum of \(L^2(\Gamma\backslash G)\); the rest of the spectrum is built from Eisenstein series induced from cusp forms on the groups \(M\). Harish-Chandra’s 1968 notes, Automorphic Forms on Semisimple Lie Groups, presented Langlands’ theory of Eisenstein series, then circulated only in manuscript.

He saw an exact parallel with Lesson 08: cusp forms play the part of the discrete series, and Eisenstein series play the part of induced representations. He called this guiding analogy the philosophy of cusp forms.

Cusp forms over every field

For the general reader. From the late 1960s Harish-Chandra worked on groups of matrices over the \(p\)-adic numbers and over finite fields. He followed what he called a Lefschetz principle: whatever is true for real groups should have an analogue for \(p\)-adic and finite groups. The building blocks in every setting are “cuspidal” representations, which cannot be obtained from smaller groups.

Finite groups of Lie type

Let \(G=\mathrm{GL}_n(\mathbb F_q)\) and let \(P=LU\) be a parabolic subgroup with Levi factor \(L\cong\mathrm{GL}_{n_1}\times\cdots\times\mathrm{GL}_{n_r}\). Harish-Chandra induction sends a representation \(\sigma\) of \(L\) to \(\operatorname{Ind}_P^G(\sigma\otimes1)\). An irreducible representation \(\pi\) of \(G\) is cuspidal if it has no non-zero vectors fixed by the unipotent radical \(U\) of any proper parabolic, the finite analogue of vanishing constant terms.

Harish-Chandra proved this using a finite analogue of his theory of Eisenstein series and constant terms. The same structure underlies the Bruhat decomposition \(G=\bigsqcup_{w\in W}BwB\), which Harish-Chandra proved for real semisimple Lie groups in 1957.

\(p\)-adic groups

For a reductive group over a \(p\)-adic field \(F\), a smooth irreducible representation is supercuspidal if its matrix coefficients are compactly supported modulo the centre. Supercuspidals are the \(p\)-adic counterpart of the discrete series, and every irreducible representation embeds in one induced from a supercuspidal of a Levi subgroup (a theorem of Jacquet). Harish-Chandra showed that admissible representations of \(p\)-adic groups in characteristic 0 have characters that are locally integrable functions, proved local expansions of characters near the identity in terms of Fourier transforms of nilpotent orbital integrals, and obtained the Plancherel formula for \(p\)-adic groups. His proof of the Plancherel formula was later written up by Jean-Loup Waldspurger (2003).

Columbia and the Institute, 1950–1983

Harish-Chandra joined Columbia University in 1950. It was the only teaching post he ever held, and his thirteen years there were his most productive. He spent 1952–53 at the Tata Institute of Fundamental Research in Bombay, founded by Bhabha, and in 1952 married Lalitha (Lily) Kale; they had two daughters. He was a member of the Institute for Advanced Study in 1955–56 and spent 1957–58 in Paris on a Guggenheim Fellowship, in close contact with André Weil. A Sloan Fellowship brought him back to the Institute in 1961–62.

His only joint paper in mathematics, with Armand Borel, appeared in 1962. In 1963 he became a permanent member of the Institute, and in 1968 its IBM–von Neumann Professor. There he finished the discrete series, the \(p\)-adic theory and the Plancherel formula. His papers were long and complete, and he released results only when the whole theory was in place.

His heart was weak. He suffered heart attacks in 1969, 1970 and 1982. In October 1983 the Institute held a conference in honour of Armand Borel’s sixtieth birthday. Shortly after it ended, on 16 October 1983, Harish-Chandra died of a heart attack while on his customary walk in Princeton. He was sixty. His collected papers, edited by V. S. Varadarajan, appeared in four volumes in 1984.

Robert Langlands, who knew him at the Institute, wrote the memoir for the Royal Society. He described a man of erect and austere bearing, with an intense inner concentration and a quick, often mischievous smile.

Honours and legacy

Harish-Chandra received the Cole Prize of the American Mathematical Society in 1954 for his papers on representations of semisimple Lie algebras and groups, especially the 1951 Transactions paper. He was an invited speaker at the International Congresses of 1954 and 1966 and gave the AMS Colloquium Lectures in 1969. He was elected a Fellow of the Royal Society in 1973, received the Srinivasa Ramanujan Medal of the Indian National Science Academy in 1974, was elected to the Indian Academy of Sciences and the Indian National Science Academy in 1975, received the Padma Bhushan in 1977, and was elected to the US National Academy of Sciences in 1981. Delhi University (1973) and Yale (1981) gave him honorary degrees. According to Langlands, he was considered for a Fields Medal in 1958. The Harish-Chandra Research Institute in Allahabad (Prayagraj) bears his name.

Legacy

• Representation theory: admissible representations, Harish-Chandra modules, infinitesimal characters, distribution characters and the discrete series are the foundation of the subject. The Langlands classification of irreducible representations of real groups (1973) is built on his tempered representations.
• Algebra: the Harish-Chandra isomorphism is the starting point of category \(\mathcal O\), of the Kazhdan–Lusztig conjectures, and of the Beilinson–Bernstein localization theorem.
• Number theory: the Borel–Harish-Chandra theorem, the finiteness theorem and the philosophy of cusp forms are the analytic ground of the Langlands programme. James Arthur’s trace formula uses his harmonic analysis throughout.
• Finite and \(p\)-adic groups: Harish-Chandra induction and cuspidal support organize the representation theory of finite groups of Lie type, developed by George Lusztig and others; his \(p\)-adic theory underlies the local Langlands correspondence.
• Physics: the Harish-Chandra–Itzykson–Zuber integral over unitary groups, from his work on the Fourier transform on Lie algebras, is a basic tool in random matrix theory.

Connections

People linked to Harish-Chandra across this collection. A name without a link has no page here yet. See Harish-Chandra in the network of all 337 lives →

Teachers and advisors

• Paul Dirac 1945-1947

Influenced by

• Atle Selberg
• Claude Chevalley
• Emil Artin
• Hermann Weyl
• Oscar Zariski 1949-1950
• K. S. Krishnan

Influenced

• Robert Langlands
• V. S. Varadarajan

Colleagues and collaborators

• André Weil 1950-1983
• Atle Selberg
• Armand Borel 1962
• Homi Bhabha 1943-1945

Friends

• Wolfgang Pauli

Also mentioned in this life

• Michael Atiyah
• Augustin-Louis Cauchy
• Joseph Fourier
• Israel Gelfand
• Charles Hermite
• David Hilbert
• Solomon Lefschetz
• Sophus Lie
• Hendrik Antoon Lorentz
• Hermann Minkowski
• Henri Poincaré
• Carl Ludwig Siegel
• John von Neumann

Fields

• representation theory 8 lives
• harmonic analysis 3 lives
• lie groups 4 lives
• automorphic forms 4 lives
• mathematical physics 45 lives

On CoursesHub

• Lie Groups & Algebras · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Harish-Chandra, MacTutor History of Mathematics.
2. Harish-Chandra, Wikipedia.
3. R. P. Langlands, Harish-Chandra 1923–1983, Biographical Memoirs of Fellows of the Royal Society 31 (1985).
4. National Academy of Sciences, Harish-Chandra, Biographical Memoir.
5. Harish-Chandra, Dictionary of Scientific Biography entry, via MacTutor.
6. Mathematics Genealogy Project, Harish-Chandra.

Mathematics

1. Harish-Chandra, Infinite irreducible representations of the Lorentz group, Proc. Roy. Soc. A 189 (1947).
2. Harish-Chandra, On some applications of the universal enveloping algebra of a semisimple Lie algebra, Trans. AMS 70 (1951).
3. Harish-Chandra, Spherical functions on a semisimple Lie group I, II, Amer. J. Math. 80 (1958).
4. A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Annals of Math. 75 (1962).
5. Harish-Chandra, Discrete series for semisimple Lie groups I, II, Acta Math. 113 (1965), 116 (1966).
6. Harish-Chandra, Automorphic Forms on Semisimple Lie Groups, notes by J. G. M. Mars, Lecture Notes in Math. 62, Springer, 1968.
7. Harish-Chandra, Harmonic Analysis on Reductive p-adic Groups, notes by G. van Dijk, Lecture Notes in Math. 162, Springer, 1970.
8. Harish-Chandra, Harmonic analysis on real reductive groups I–III, J. Funct. Anal. 19 (1975); Invent. Math. 36 (1976); Annals of Math. 104 (1976).
9. Harish-Chandra, Collected Papers, 4 vols., ed. V. S. Varadarajan, Springer, 1984.
10. V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals of Math. 48 (1947).
11. S. Lang, SL₂(R), Addison-Wesley, 1975; Springer GTM 105.
12. A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986.
13. S. Helgason, Groups and Geometric Analysis, Academic Press, 1984.
14. J.-L. Waldspurger, La formule de Plancherel pour les groupes p-adiques, d’après Harish-Chandra, J. Inst. Math. Jussieu 2 (2003).