Continuous series of representations

principal series of representations

The family of irreducible unitary representations of a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c0257501.png" /> that occur in the decomposition of the regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c0257502.png" />, but do not belong to the discrete series (of representations) of this group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c0257503.png" /> is a real semi-simple Lie group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c0257504.png" /> is its Iwasawa decomposition and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c0257505.png" /> is the centralizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c0257506.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c0257507.png" />, then the non-degenerate continuous principal series of representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c0257508.png" /> is the family of irreducible unitary representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c0257509.png" /> induced by the finite-dimensional irreducible unitary representations of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575010.png" /> that are trivial on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575011.png" />.

The complementary (or degenerate) continuous series of representations of such a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575012.png" /> is the family of irreducible unitary representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575013.png" /> that occur in the complementary series (of representations) (respectively, the degenerate series of representations) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575014.png" /> and are not isolated points of it (as subsets of the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575015.png" />). The analytic continuation of the non-degenerate continuous principal series of representations is the family of (generally speaking, non-unitary) representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575016.png" /> induced by all possible finite-dimensional irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575017.png" /> that are trivial on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575018.png" />. This family plays a decisive role in the representation theory of real semi-simple Lie algebras and harmonic analysis on these groups; in particular, any completely irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575019.png" /> in a Hilbert space is infinitesimally equivalent to a subrepresentation of some quotient representation of one of the representations of the analytic continuation of the non-degenerate continuous principal series of representations. See also Infinite-dimensional representation of a Lie group.

References

[1]	I.M. Gel'fand, M.A. Naimark, "Unitäre Darstellungen der klassischen Gruppen" , Akademie Verlag (1957) (Translated from Russian)
[2]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[3]	G. Warner, "Harmonic analysis on semi-simple Lie groups" , 1–2 , Springer (1972)

Comments

In the Western literature one customarily uses the terminology principal series instead of non-degenerate continuous principal series. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575020.png" /> as above is a minimal parabolic subgroup. One even speaks of a (generalized) principal series representation in the following situation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575021.png" /> be a connected semi-simple matrix group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575022.png" /> be a cuspidal (not necessarily minimal) parabolic subgroup. Fix a Langlands decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575023.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575024.png" /> the unipotent radical and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575025.png" /> a vector group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575026.png" /> be a discrete series representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575028.png" /> a (non-unitary) character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575029.png" />. The induced representation

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575030.png" />

is called a generalized principal series representation. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575031.png" /> is unitary, these are the representations occurring in Harish-Chandra's Plancherel formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575032.png" />. The result mentioned at the end of the entry above usually is called the subquotient theorem: Any irreducible admissible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025750/c02575033.png" /> can be realized canonically as a subquotient of a generalized principal series representation.

References

[a1]	A.W. Knopp, "Representation of semisimple groups" , Princeton Univ. Press (1986)
[a2]	D.A. Vogan jr., "Representations of real reductive Lie groups" , Birkhäuser (1981)