It has been 10 days since the Shah of Iran concluded his spectacular Washington visit. He left behind a battleground of litter that cost $20,000 to clean and military requests that will cost nearly $8 million to fill.

He also left behind two quiet investigations - one by the Justice Department, the other by a Senate Foreign Relations subommittee. The investigators want to know if the Shah's supporters, who were flown by the planeload, violated federal law by failing to register as foreign agents.

It might be more instructive, however, if the investigators could determine the source of the Shah's remarkable influence in Washington. No other sovereign customer has wangled so much military equipment out of the United States. Now he has asked the Pentagon for additional jet fighters, radar planes, military transport planes and patrol boats.

All this has antagonized Iran's neighbor, Saudi Arabia, whose oil has become essential to the well being of all Americans. Yet Washington invariably has favored the Shah over the more reliable Saudis. What is all the more amazing, he has succeeded without the backing of an ethnic pressure group such as the one supporting Israel or an "Iran Lobby."

Under Democrats and Republicans alike, the White House has pampered the Shah, the Pentagon has attended to his wants and the State Department has been devoted to him. As for the Centrial Intelligence Agency, it has positively adored him.

We have conducted an investigation, for example, of the Shah's association with ex-CIA director Richard Helms. It began in Switzerland where the Shah and Helms attended LeRosey preparatory school. The Shah was then a weak, whining prince bullied by a cruel, domineering father; Helms was an elegant-mannered son of aristocrats.

During World War II, the edler Shah demonstrated an unfortunate preference for the Nazis. The Allies duly deposed him and deposited his 21-year-old son on the throne. Helms, meanwhile, became a top official of the CIA, which in 1953 arranged a coup against Iranian Premier Mohammed Mossadegh and saved the Shah's throne.

Helms belonged to the sophisticated, old-school crowd who ran the CIA. He became the CIA director about the time that a colleague, Joseph Farland, became ambassador to Pakistan. At Islamabd, Farland met almost daily with Gen. Yahya Khan, then the military dictator. Through him, Farland helped Henry Kissinger through the Bamboo Curtain and provided a cover for Kissinger's secret, historic visit to Peking.

After Yahya's government collapsed, Farland was assigned to be ambassador to Iran. He had scarcely settled down at his new port when Helms suddenly found himself in need of a refuge from prying Senate investigators. It isn't everyone who will accept a CIA chief as an ambassador. But the Shah was glad to welcome his former schoolmate.

Witnesses have told us that Farland was "greatly amazed" and "grief-stricken" at being removed. His abrupt departre was so emotional, according to our sources, that he was moved to tears.

Once installed in Iran, Helms behaved as if he were representing the Shah, rather than the American people. In 1973, for example, the Shah was preparing for a trip to the United States. He hinted to Helms that he did not wish to be dodged by demonstrators wearing masks and waving signs. Helms dutifully fired off a classified cable to the State Department relaying the Shah's suggestion "that we tidy up as much as possible anti-Shah element in U.S."

Helms also asked the State Department to determine whether an anti-Shah newspaper in Washington could be closed down. The deparment's lawyers researched the question and determinded that the publsiher, Nasser Afshar, was an American citizen protected by the First Amendment.

The lawyers advised Helms that the mailing of the newspaper to the embassy could be stopped if the recipients wrote "Refuse to accept" on the envelopes. Embassy sources report that Helms, consulting no one, made the unilateral decision that none of his employees should read the offending newspaper. Thereafter all copies were returned unopened, each identically stamped: "Refused by addressee Return to sender."

The Shah regulated in other ways what Helms's employees could read about the Shah. In 1975-76, for example, the CIA's classified background reports on repression in Iran became increasingly critical. Suddenly, distribution was cut off to embassy officials who had been reading them regularly.

In our July 11, 1975, column, we cited an unflattering CIA psyhological profile of the Shah, portraying him as a brilliant but dangerous megalomaniac. Helms "exploded" when he read the column, according to sources who were present. Thereafter he would not allow the column to be circulated among the staff.

Helms finally resigned as ambassador to Iran on Jan. 31, 1977. He told colleagues at the embassy that he was leaving "to make some money." He returned to Washington to take his lumps for lying to the Senate earlier about the CIA's involvement in Chilean politics.

L ast month, he quitely opened a "consulting" firm in Washington for the purpose, he told us, of acting as a "go-between" for foreign interests seeking to do business in the United States and vice versa. He calls his new firm the "Safeer" company. That's Persian for "ambassador."

There's something here that smells like a "mahi." That's Persian for "fish."

Footnote: Still the charming patrician, Helms told us he is now in "private life" and has "no relationship" with the Iranian government. His lawyers had advised him, therefore, that his new firm was not in conflict with federal law, he said. He hasn't registered as a foreign agent, he added, because he doesn't "intend to do any lobbying." He refused to discuss his actions as ambassador to Iran.