In large Tokyo newspaper advertisement, the cheerful face of Alex Haley beams across a table on which rests a bottle of Very Rare Old Suntory whiskey.

He is explaining the popularity of his famous book, "Roots." People worry about the disruption of modern family life, he says, and there is nostalgia for old ways of living. Many old things, such as antiques, are popular all over the world.

Then comes the sales pitch: "I am one of the fans of Suntory Old."

The ad is testimony of the phenomenal popularity in Japan this fall of Haley's epic story of black survival and of his own quest for family origins.

His book is being published here in two volumes and the first is already selling briskly. Last week, the television show, with Japanese dubbings, was serialized in eight parts on prime time. It produced ratings that delighted the commercial network.

Haley and the word "Roots" itself have been firmly established almost overnight in Japanese popular culture. They are the fall season's American-import fad in a country that quickly embraces and absorbs any trend that arouses mass appeal in the United States.

A Japanese movie about an ancient family curse is advertised as "a tragedy of roots." In a subway display a copy of Haley's book is nestled beside an open bottle of Dewar's scotch. Such phrases as "the roots of the rumors are . . ." suddenly crop up in conversations. Inexplicably, the word has been borrowed to described a play in the popular game of Mah-Jongg.

A refrigerator company ad on television depicts how 200 years ago, ice was carved from Mount Fuji and carried to the homes of the shoguns who ruled Japan. That, it declared, was the "roots" of refrigeration.

The Japanese offer a number of explanations for the popularity of "Roots." For one thing it is another piece of exotic Americana - like T-shirts lettered with the names of obscure colleges - that entrances Asians.

Beyond that however, a number of observes familiar with the saga of Kunta Kinte and his descendants believe it has struck a deeper chord here because Japanese are forever concerned with discovering their own roots.

Tadao Kawamura, the Japanese editor whose company is publishing "Roots," was first attracted to the story when he read an article about Haley in Reader's Digest. The search for ancestors, he said recently, "is a feeling that Japanese can share." They are also aroused by stories of the problems of colored races, he said. His expectations have been satisfied by the rapid sale of 230,000 copies of the first half of the book in Japanese.

To Masao Kunihiro a cultural authropologist the popularity of the book and television serial is a reflection of when what he calls a new wave of "cultural nationalism" washing over Japan.

A vast academic literature has been published recently about the origins of the Japanese and their language. Scholars debate where the first immigrants came from in ancient times. Even in the popular media, Kunihiro observed there is a keen interest in Japan's genesis, and Haley's search for his own family's origins in Africa is a familiar theme.

His opinion is underscored by Nachiro Nakamura who as director of the international department of the Asahi National Broadcastings Co., was primarily responsible for bringing "Roots" to Japanese television.

Japanese, he said [WORD ILLEGIBLE] sensitive these days to their own less of traditions. Old wars of doing things are disappering and many Japanese miss them. "Roots" helps them recall their own past, Nakamura offers this example:

Young Japanese once were taught to respect their fathers and teachers, who were stern and exacting people of great dignity. The respect is fading particularly among young boys. On the television screen, a most memorable scene is the one in which Kunta Kinte and his friends are taught to be warriors by a demanding but dignified African elder. The Japanese watch and remember their own elders.

For Nakamura and other Asahi executives, the serialization of "Roots" was a long and - for a time - dubious project.

Alerted by their New York correspondent to the extraordinary impact of the show in the United States, they at first doubted that any story of black Americans could become popular here. Cowboys and cops have been television fare for years in Japan, but this was something different.

"Japanese audiences usually prefer something about white people," in foreign films. Nakamura said, "and we were not very interested at first." There was also an early technical mishap: The Asahi correspondent was unable to make a videotape of the final American episode because his television antenna blew down in a New York snowstorm.

When Nakamura and other Asahi officials saw the complete series, however, they agreed to take a chance and buy it from Warner Brothers. Nakamura has had experience in risky ventures - last spring he produced an expurgated version of the French pornographic film "Emanuelle."

It was the "high quality" of the dramatization in "Roots" that finally sold the people at Asahi. Nakamura said. Despite the serious often tragic theme, he said, the studio felt that the dramatization was "not too artistic, not too high-brow" for Japanese audiences.

Asahi Broadcasting was pleasantly surprised at the ratings for the "Roots" series and is making plans for a second showing this winter.

One particularly pleasing result was the show's attraction for younger men who hardly find much in the typical prime-time situation comedy to interest them, and who would rather spend their evenings on the town.

"A lot of nightclubs and bars and pinball parlors lost their customers that week," said Nakamura.

one explaination seems to be the heavy publicity "Roots" received in sex magazines aimed at Japanese men. Extracts from the book are printed in one recent issue of the Japanese Playboy, along with the photographs of nude women.